%%Page: 1 1 1 0 bop Black 0 TeXcolorgray Black Black Black Black 750 755 a @beginspecial 0 @llx 0 @lly 99 @urx 16 @ury 2880 @rwi @setspecial%%BeginDocument: logo21.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: logo21.eps %%Creator: fig2dev Version 3.2.3 Patchlevel %%CreationDate: Thu Sep 13 16:47:41 2001 %%For: pope@fry.research.att.com (Sue Pope) %%BoundingBox: 0 0 99 16 %%Magnification: 0.1500 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 16 moveto 0 0 lineto 99 0 lineto 99 16 lineto closepath clip newpath -12.0 26.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin %%Page: 1 1 10 setmiterlimit 0.00900 0.00900 sc 7.500 slw % Ellipse n 2213 1988 856 856 0 360 DrawEllipse gs col4 1.00 shd ef gr gs col4 s gr % Ellipse n 2213 1991 813 813 0 360 DrawEllipse gs col8 1.00 shd ef gr gs col8 s gr % Ellipse n 1602 1879 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2001 2575 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2831 1877 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2618 1509 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 1806 1519 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2205 1364 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 1669 2290 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2762 2305 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2439 2579 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2220 1992 440 440 0 360 DrawEllipse gs col4 1.00 shd ef gr gs col4 s gr % Ellipse n 2618 1509 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 1805 1515 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 1669 2290 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 2205 1360 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 2762 2305 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 1600 1875 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 2439 2579 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 2831 1877 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 2001 2573 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr /Times-Bold ff 225.00 scf sf 1997 2652 m gs 1 -1 sc (23) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Bold ff 225.00 scf sf 2438 2659 m gs 1 -1 sc (11) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 480.00 scf sf 3375 2205 m gs 1 -1 sc (Article 01.2.1) col0 sh gr /Times-Roman ff 480.00 scf sf 3375 1650 m gs 1 -1 sc (Journal of Integer Sequences, Vol. 4 \(2001\),) col0 sh gr /Times-Bold ff 225.00 scf sf 2618 1592 m gs 1 -1 sc (2) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Bold ff 225.00 scf sf 2830 1956 m gs 1 -1 sc (3) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Bold ff 225.00 scf sf 2761 2386 m gs 1 -1 sc (6) dup sw pop 2 div neg 0 rm col0 sh gr % Ellipse n 1805 1517 42 42 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Polyline n 2554 1797 m 2554 1796 l 2554 1790 l 2554 1779 l 2554 1765 l 2553 1753 l 2552 1743 l 2550 1736 l 2548 1730 l 2544 1725 l 2540 1721 l 2534 1717 l 2528 1713 l 2521 1711 l 2514 1709 l 2507 1708 l 2499 1707 l 2492 1707 l 2483 1707 l 2474 1707 l 2465 1708 l 2456 1710 l 2447 1712 l 2438 1714 l 2431 1717 l 2423 1720 l 2416 1724 l 2408 1728 l 2401 1734 l 2394 1739 l 2388 1745 l 2382 1751 l 2377 1756 l 2372 1762 l 2368 1768 l 2363 1775 l 2359 1783 l 2356 1791 l 2352 1799 l 2350 1807 l 2348 1814 l 2346 1823 l 2345 1831 l 2345 1841 l 2345 1852 l 2346 1863 l 2349 1873 l 2352 1884 l 2356 1894 l 2360 1902 l 2365 1910 l 2371 1919 l 2378 1928 l 2385 1937 l 2393 1946 l 2401 1955 l 2409 1963 l 2417 1970 l 2425 1977 l 2435 1985 l 2446 1993 l 2457 2000 l 2467 2008 l 2477 2015 l 2487 2021 l 2494 2027 l 2501 2033 l 2506 2038 l 2510 2043 l 2513 2049 l 2515 2055 l 2516 2061 l 2516 2067 l 2516 2073 l 2514 2079 l 2510 2087 l 2505 2095 l 2498 2104 l 2490 2113 l 2481 2120 l 2473 2125 l 2465 2129 l 2458 2131 l 2449 2133 l 2440 2134 l 2431 2134 l 2422 2133 l 2414 2131 l 2406 2128 l 2396 2122 l 2385 2115 l 2375 2107 l 2366 2101 l 2359 2097 l 2353 2096 l 2350 2098 l 2347 2101 l 2346 2107 l 2344 2114 l 2344 2122 l 2344 2130 l 2345 2138 l 2346 2145 l 2347 2151 l 2349 2158 l 2352 2164 l 2355 2169 l 2360 2174 l 2364 2179 l 2370 2182 l 2375 2186 l 2382 2188 l 2390 2190 l 2399 2193 l 2409 2194 l 2419 2195 l 2429 2196 l 2438 2196 l 2448 2196 l 2456 2195 l 2465 2194 l 2475 2192 l 2484 2189 l 2493 2187 l 2501 2183 l 2509 2180 l 2516 2177 l 2524 2172 l 2533 2166 l 2541 2159 l 2548 2152 l 2555 2144 l 2560 2137 l 2565 2129 l 2569 2121 l 2573 2111 l 2576 2102 l 2579 2093 l 2581 2084 l 2583 2076 l 2584 2068 l 2584 2058 l 2584 2049 l 2583 2040 l 2582 2032 l 2580 2023 l 2577 2014 l 2572 2004 l 2568 1994 l 2563 1984 l 2558 1976 l 2552 1968 l 2546 1960 l 2538 1952 l 2531 1945 l 2524 1938 l 2517 1932 l 2509 1926 l 2501 1921 l 2493 1914 l 2484 1908 l 2476 1902 l 2468 1897 l 2461 1890 l 2453 1883 l 2445 1876 l 2437 1869 l 2431 1862 l 2427 1856 l 2423 1849 l 2420 1842 l 2418 1835 l 2417 1830 l 2417 1825 l 2417 1820 l 2417 1816 l 2418 1812 l 2419 1808 l 2419 1806 l 2420 1803 l 2421 1801 l 2422 1798 l 2424 1796 l 2425 1794 l 2427 1792 l 2429 1789 l 2431 1787 l 2433 1785 l 2436 1783 l 2437 1781 l 2439 1780 l 2442 1779 l 2444 1778 l 2446 1778 l 2448 1777 l 2450 1777 l 2452 1777 l 2455 1777 l 2457 1777 l 2460 1776 l 2462 1776 l 2465 1775 l 2468 1775 l 2470 1775 l 2473 1775 l 2476 1775 l 2479 1776 l 2482 1776 l 2484 1777 l 2487 1777 l 2489 1777 l 2492 1778 l 2495 1779 l 2498 1780 l 2502 1782 l 2506 1785 l 2510 1788 l 2514 1790 l 2517 1792 l 2520 1794 l 2522 1795 l 2524 1796 l 2525 1798 l 2527 1799 l 2529 1800 l 2531 1802 l 2534 1805 l 2536 1807 l 2539 1809 l 2540 1811 l 2542 1812 l 2544 1813 l 2546 1814 l 2547 1814 l 2549 1813 l 2550 1811 l 2552 1808 l 2554 1804 l 2555 1801 l 2556 1798 l 2556 1796 l 2557 1795 l 2557 1793 l 2557 1792 l gs 0.00 setgray ef gr gs col0 s gr % Polyline n 1871 1753 m 1902 1705 l 2090 1705 l 2075 1753 l 2075 2255 l 2012 2317 l 1933 2317 l 2012 2255 l 2012 1753 l 1871 1753 l 1918 1737 l cp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 2169 1705 m 2278 1705 l 2263 1720 l 2247 1753 l 2247 2129 l 2278 2191 l 2152 2191 l 2169 2176 l 2185 2129 l 2185 2098 l 2185 1753 l cp gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 1600 1877 42 42 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr /Times-Bold ff 225.00 scf sf 2197 1439 m gs 1 -1 sc (1) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Bold ff 225.00 scf sf 1667 2376 m gs 1 -1 sc (47) dup sw pop 2 div neg 0 rm col0 sh gr $F2psEnd rs %%EndDocument @endspecial 487 1119 a Fz(Arithmetic)66 b(and)e(gro)-5 b(wth)64 b(of)h(p)5 b(erio)g(dic)1705 1280 y(orbits)1200 1776 y Fy(Y)-10 b(ash)39 b(Puri)p 0 .5 0 TeXcolorrgb -46 x Fx(1)p (#Hfootnote.1) [[277 504 281 516] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1743 1776 a Fy(and)g(Thomas)e(W)-10 b(ard)666 1969 y(Sc)m(ho)s(ol)37 b(of)i(Mathematics,)d(Univ)m(ersit)m(y)g(of)i(East)g (Anglia,)1324 2110 y(Norwic)m(h)f(NR4)h(7TJ,)g(U.K.)1031 2354 y(Email)f(addresses:)p 0 1 0 0 TeXcolorcmyk 51 w(t.w)m (ard@uea.ac.uk)p Black [[304 435 412 447] [1 1 1 [3 3]] [0 0 1]] (mailto:t.ward@uea.ac.uk) pdfm 1693 3086 a Fw(Abstract)p 0 .5 0 TeXcolorrgb Black 456 3245 a Fv(Two)43 b(natur)-5 b(al)45 b(pr)-5 b(op)g(erties)43 b(of)h(inte)-5 b(ger)44 b(se)-5 b(quenc)g(es)44 b(ar)-5 b(e)44 b(intr)-5 b(o)g(duc)g(e)g(d)43 b(and)h(stud-)456 3361 y(ie)-5 b(d.)85 b(The)48 b(\014rst,)k Fu(exact)c(realizabilit)m(y) p Fv(,)53 b(is)c(the)f(pr)-5 b(op)g(erty)49 b(that)g(the)f(se)-5 b(quenc)g(e)456 3477 y(c)g(oincides)34 b(with)h(the)h(numb)-5 b(er)35 b(of)g(p)-5 b(erio)g(dic)35 b(p)-5 b(oints)35 b(under)g(some)g(map.)46 b(This)35 b(is)456 3593 y(shown)30 b(to)i(imp)-5 b(ose)30 b(a)i(str)-5 b(ong)31 b(inner)g(structur)-5 b(e)32 b(on)f(the)h(se)-5 b(quenc)g(e.)43 b(The)31 b(se)-5 b(c)g(ond,)456 3710 y Fu(realizabilit)m(y)40 b(in)g(rate)p Fv(,)j(is)d(the)i(pr)-5 b(op)g(erty)41 b(that)g(the)h(se)-5 b(quenc)g(e)40 b(asymp)-5 b(ototic)g(al)5 b(ly)456 3826 y(appr)-5 b(oximates)37 b(the)h(numb)-5 b(er)38 b(of)g(p)-5 b(erio)g(dic)37 b(p)-5 b(oints)38 b(under)h(some)e(map.)55 b(In)38 b(b)-5 b(oth)456 3942 y(c)g(ases)42 b(we)h(discuss)g(when)g(a)g (se)-5 b(quenc)g(e)43 b(c)-5 b(an)43 b(have)g(that)g(pr)-5 b(op)g(erty.)71 b(F)-7 b(or)43 b(exact)456 4058 y(r)-5 b(e)g(alizability,)39 b(this)f(amounts)g(to)h(examining)e(the)i(r)-5 b(ange)38 b(and)g(domain)f(among)456 4175 y(inte)-5 b(ger)34 b(se)-5 b(quenc)g(es)34 b(of)h(the)f(p)-5 b(air)g(e)g(d)35 b(tr)-5 b(ansformations)521 4419 y(Per)673 4434 y Ft(n)747 4419 y Fs(=)851 4324 y Fr(X)874 4540 y Ft(d)p Fq(j)p Ft(n)1012 4419 y Fp(d)34 b Fv(Orb)1258 4434 y Ft(d)1298 4419 y Fs(;)316 b Fv(Orb)1801 4434 y Ft(d)1870 4419 y Fs(=)1988 4351 y(1)p 1983 4396 59 4 v 1983 4487 a Fp(n)2068 4324 y Fr(X)2090 4540 y Ft(d)p Fq(j)p Ft(n)2228 4419 y Fp(\026)p Fs(\()p Fp(n=d)p Fs(\))p Fv(Per)2673 4434 y Ft(d)3012 4419 y Fo(ORBIT)456 4729 y Fv(that)38 b(move)f(b)-5 b(etwe)g(en)38 b(an)f(arbitr)-5 b(ary)39 b(se)-5 b(quenc)g(e)37 b(of)h(non-ne)-5 b(gative)36 b(inte)-5 b(gers)38 b(Orb)456 4845 y(c)-5 b(ounting)41 b(the)h(orbits)f(of)h(a)f(map)g(and)g(the)h (se)-5 b(quenc)g(e)41 b(Per)h(of)f(p)-5 b(erio)g(dic)41 b(p)-5 b(oints)p Black 456 5023 499 4 v 555 5084 a Fx(1)p 0 TeXcolorgray Black 32 x Fn(The)51 b(\014rst)g(author)f(gratefully)g (ac)n(kno)n(wledges)e(the)k(supp)r(ort)e(of)h(E.P)-7 b(.S.R.C.)51 b(gran)n(t)456 5216 y(96001638)p Black Black 1931 5315 a Fm(1)p Black eop %%Page: 2 2 2 1 bop Black 0 TeXcolorgray Black 456 251 a Fm(2)p Black 456 450 a Fv(for)41 b(that)h(map.)63 b(Sever)-5 b(al)41 b(examples)f(fr)-5 b(om)41 b(the)p 0 1 0 0 TeXcolorcmyk 41 w(Encyclop)-5 b(e)g(dia)40 b(of)h(Inte)-5 b(ger)41 b(Se-)p [[344 663 485 675] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/~njas/sequences/) pdfm 456 566 a(quenc)-5 b(es)p Black [[72 649 165 661] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/~njas/sequences/) pdfm 34 w(arise)35 b(in)g(this)g(work,)g(and)g(a)g(table)g(of)g(se)-5 b(quenc)g(es)34 b(fr)-5 b(om)35 b(the)g(Encyclo-)456 683 y(p)-5 b(e)g(dia)34 b(known)g(or)g(c)-5 b(onje)g(ctur)g(e)g(d)35 b(to)g(b)-5 b(e)34 b(exactly)h(r)-5 b(e)g(alizable)34 b(is)g(given.)1549 911 y Fs(1.)49 b Fl(Intr)n(oduction)555 1085 y Fs(Let)34 b Fp(T)42 b Fs(:)29 b Fp(X)36 b Fk(!)29 b Fp(X)40 b Fs(b)s(e)34 b(a)f(map.)45 b(Three)35 b(measures)f(of)f(gro) m(wth)h(in)f(complexit)m(y)456 1202 y(for)f Fp(T)46 b Fs(are)32 b(giv)m(en)i(b)m(y)g(the)f Fu(n)m(um)m(b)s(er)h(of)e(p)s(oin) m(ts)h(with)g(p)s(erio)s(d)g Fp(n)p Fs(,)1305 1369 y Fp(f)1353 1384 y Ft(n)1400 1369 y Fs(\()p Fp(T)14 b Fs(\))27 b(=)h(#)p Fk(f)p Fp(x)g Fk(2)g Fp(X)36 b Fk(j)27 b Fp(T)2229 1328 y Ft(n)2276 1369 y Fp(x)h Fs(=)f Fp(x)p Fk(g)p Fp(;)456 1537 y Fs(the)33 b Fu(n)m(um)m(b)s(er)h(of)e(p)s(oin)m(ts)h(with)g (least)g(p)s(erio)s(d)g Fp(n)p Fs(,)818 1705 y Fp(f)877 1663 y Fq(\003)866 1729 y Ft(n)916 1705 y Fs(\()p Fp(T)14 b Fs(\))27 b(=)h(#)p Fk(f)p Fp(x)g Fk(2)g Fp(X)36 b Fk(j)27 b Fp(T)1745 1663 y Ft(n)1792 1705 y Fs(\()p Fp(x)p Fs(\))h(=)f Fp(x)33 b Fs(and)g(#)p Fk(f)p Fp(T)2534 1663 y Ft(k)2576 1705 y Fp(x)p Fk(g)2681 1720 y Ft(k)r Fq(2)p Fj(Z)2843 1705 y Fs(=)28 b Fp(n)p Fk(g)p Fp(;)456 1872 y Fs(and)k(the)h Fu(n)m(um)m(b)s(er)h(of)e(orbits)h(of)f(length)h Fp(n)p Fs(,)1573 2040 y Fp(f)1632 1999 y Ft(o)1621 2065 y(n)1670 2040 y Fs(\()p Fp(T)14 b Fs(\))27 b(=)h Fp(f)2007 1999 y Fq(\003)1996 2065 y Ft(n)2046 2040 y Fs(\()p Fp(T)14 b Fs(\))p Fp(=n:)456 2208 y Fs(In)31 b(this)g(note)g(w)m(e)h(assume)h (that)d Fp(f)1714 2223 y Ft(n)1762 2208 y Fs(\()p Fp(T)14 b Fs(\))30 b(is)h(\014nite)h(for)e Fp(n)e Fk(\025)g Fs(1)j(and)g(giv)m (e)h(some)g(re-)456 2324 y(sults)44 b(on)f(what)h(arithmetic)h(prop)s (erties)f(the)g(sequence)i(\()p Fp(f)2728 2339 y Ft(n)2775 2324 y Fs(\()p Fp(T)14 b Fs(\)\))43 b(ma)m(y)h(ha)m(v)m(e,)456 2440 y(and)51 b(sho)m(w)h(when)h(the)e(gro)m(wth)h(in)f(\()p Fp(f)1949 2455 y Ft(n)1996 2440 y Fs(\()p Fp(T)14 b Fs(\)\))51 b(is)g(related)h(to)f(the)h(gro)m(wth)f(in)456 2556 y(\()p Fp(f)553 2520 y Fq(\003)542 2581 y Ft(n)592 2556 y Fs(\()p Fp(T)14 b Fs(\)\).)63 b(It)39 b(will)h(b)s(e)f(con)m(v)m(enien)m(t)j (to)d(adopt)g(the)h(follo)m(wing)g(notation:)56 b(a)39 b(se-)456 2673 y(quence)34 b Fp(a)829 2688 y Fi(1)869 2673 y Fp(;)17 b(a)964 2688 y Fi(2)1003 2673 y Fp(;)g(a)1098 2688 y Fi(3)1137 2673 y Fp(;)g(:)g(:)g(:)49 b Fs(is)33 b(denoted)g(\()p Fp(a)1900 2688 y Ft(n)1947 2673 y Fs(\))g(or)f(simply) i Fp(a)p Fs(.)p Black 456 2853 a Fo(De\014nition)39 b(1.1.)p Black 42 w Fs(Let)34 b Fp(\036)c Fs(=)f(\()p Fp(\036)1661 2868 y Ft(n)1708 2853 y Fs(\))k(b)s(e)h(a)g(sequence)i(of)d (non-negativ)m(e)i(in)m(tegers.)456 2969 y(Then)p Black 604 3111 a(\(1\))p Black 41 w Fp(\036)54 b Fk(2)h(E)9 b(R)48 b Fs(\()p Fu(exactly)h(realizable)p Fs(\))g(if)f(there)h(is)g(a) e(set)i Fp(X)56 b Fs(and)48 b(a)g(map)770 3227 y Fp(T)42 b Fs(:)27 b Fp(X)36 b Fk(!)27 b Fp(X)40 b Fs(for)32 b(whic)m(h)j Fp(f)1765 3242 y Ft(n)1812 3227 y Fs(\()p Fp(T)14 b Fs(\))27 b(=)g Fp(\036)2147 3242 y Ft(n)2227 3227 y Fs(for)32 b(all)g Fp(n)c Fk(\025)g Fs(1;)p Black 604 3344 a(\(2\))p Black 41 w Fp(\036)52 b Fk(2)h(RR)48 b Fs(\()p Fu(realizable)g(in)f (rate)p Fs(\))g(if)g(there)g(is)h(a)f(set)g Fp(X)55 b Fs(and)47 b(a)g(map)770 3460 y Fp(T)42 b Fs(:)27 b Fp(X)36 b Fk(!)27 b Fp(X)40 b Fs(for)32 b(whic)m(h)j Fp(f)1765 3475 y Ft(n)1812 3460 y Fs(\()p Fp(T)14 b Fs(\))p Fp(=\036)2066 3475 y Ft(n)2139 3460 y Fk(!)28 b Fs(1)k(as)h Fp(n)28 b Fk(!)f(1)p Fs(.)555 3640 y(None)38 b(of)e(the)i(results)g(b)s(elo)m (w)g(are)f(c)m(hanged)h(if)f(the)g(realizing)h(maps)g(are)f(re-)456 3757 y(quired)30 b(to)e(b)s(e)i(homeomorphisms)h(of)d(a)h(compact)g Fp(X)8 b Fs(,)30 b(but)f(this)h(is)f(not)g(pursued)456 3873 y(here.)1425 4101 y(2.)49 b Fl(Exa)n(ct)37 b(realiza)-7 b(tion)555 4276 y Fs(The)34 b(set)g(of)e(p)s(oin)m(ts)i(with)f(p)s (erio)s(d)g Fp(n)g Fs(under)h Fp(T)47 b Fs(is)33 b(the)g(disjoin)m(t)h (union)f(of)g(the)456 4392 y(set)g(of)f(p)s(oin)m(ts)h(with)g(least)h (p)s(erio)s(d)e Fp(d)g Fs(under)i Fp(T)46 b Fs(for)32 b Fp(d)g Fs(dividing)i Fp(n)p Fs(,)f(so)1547 4576 y Fp(f)1595 4591 y Ft(n)1642 4576 y Fs(\()p Fp(T)14 b Fs(\))28 b(=)1920 4481 y Fr(X)1943 4697 y Ft(d)p Fq(j)p Ft(n)2081 4576 y Fp(f)2140 4534 y Fq(\003)2129 4600 y Ft(d)2179 4576 y Fs(\()p Fp(T)14 b Fs(\))p Fp(:)967 b Fs(\(1\))456 4858 y(Equation)30 b(\()p 0 1 0 0 TeXcolorcmyk(1)p Black (#equation.1) [[181 134 187 146] [1 1 1 [3 3]] [0 0 1]] pdfm (\))e(ma) m(y)i(b)s(e)g(in)m(v)m(erted)h(via)e(the)h(M\177)-49 b(obius)30 b(in)m(v)m(ersion)h(form)m(ula)f(to)e(giv)m(e)1405 5101 y Fp(f)1464 5060 y Fq(\003)1453 5126 y Ft(n)1503 5101 y Fs(\()p Fp(T)14 b Fs(\))27 b(=)1780 5006 y Fr(X)1803 5222 y Ft(d)p Fq(j)p Ft(n)1941 5101 y Fp(\026)p Fs(\()p Fp(n=d)p Fs(\))p Fp(f)2282 5116 y Ft(d)2322 5101 y Fs(\()p Fp(T)14 b Fs(\))p Fp(;)824 b Fs(\(2\))p Black Black eop %%Page: 3 3 3 2 bop Black 0 TeXcolorgray Black 3406 251 a Fm(3)p Black 456 450 a Fs(where)28 b Fp(\026)p Fs(\()p Fk(\001)p Fs(\))e(is)i(the)f(M\177)-49 b(obius)28 b(function.)43 b(On)27 b(the)g(other)g(hand,)i(the)e(set)h(of)f(p)s(oin)m(ts)456 566 y(with)33 b(least)g(p)s(erio)s(d)f Fp(n)h Fs(comprises)i(exactly)f Fp(f)2141 530 y Ft(o)2130 591 y(n)2211 566 y Fs(orbits)f(eac)m(h)h(of)e (length)h Fp(n)p Fs(,)g(so)1068 774 y(0)27 b Fk(\024)h Fp(f)1308 733 y Fq(\003)1297 798 y Ft(n)1348 774 y Fs(\()p Fp(T)14 b Fs(\))27 b(=)1625 679 y Fr(X)1648 895 y Ft(d)p Fq(j)p Ft(n)1786 774 y Fp(\026)p Fs(\()p Fp(n=d)p Fs(\))p Fp(f)2127 789 y Ft(d)2167 774 y Fs(\()p Fp(T)14 b Fs(\))27 b Fk(\021)h Fs(0)k(mo)s(d)h Fp(n:)488 b Fs(\(3\))456 1074 y(It)28 b(is)h(clear)g(\(since)h(one)f(ma)m(y)g(tak)m(e)g Fp(X)36 b Fs(=)27 b Fh(N)43 b Fs(and)29 b(mak)m(e)h Fp(T)42 b Fs(to)28 b(b)s(e)h(a)f(p)s(erm)m(utation)456 1190 y(with)39 b(the)g(appropriate)g(n)m(um)m(b)s(er)i(of)d(cycles)j(of)d(eac)m(h)i (length\))g(that)e(these)i(are)456 1307 y(the)33 b(only)g(conditions)g (for)f(mem)m(b)s(ership)j(in)e Fk(E)9 b(R)p Fs(.)p Black 456 1498 a Fo(Lemma)41 b(2.1.)p Black 42 w Fv(L)-5 b(et)37 b Fp(\036)f Fv(b)-5 b(e)36 b(a)g(se)-5 b(quenc)g(e)36 b(of)g(non-ne)-5 b(gative)35 b(inte)-5 b(gers.)48 b(Then)36 b Fp(\036)30 b Fk(2)456 1614 y(E)9 b(R)33 b Fv(if)h(and)g(only)f(if) 1222 1539 y Fr(P)1327 1643 y Ft(d)p Fq(j)p Ft(n)1447 1614 y Fp(\026)p Fs(\()p Fp(n=d)p Fs(\))p Fp(\036)1798 1629 y Ft(d)1871 1614 y Fv(is)h(non-ne)-5 b(gative)32 b(and)i(divisible)e(by)i Fp(n)h Fv(for)456 1737 y(al)5 b(l)34 b Fp(n)28 b Fk(\025)g Fs(1)p Fv(.)555 1928 y Fs(Ev)m(erything)34 b(that)d(follo)m(ws)h(is)f(a)g(consequence)k(of)30 b(this)i(lemma.)44 b(Before)32 b(con-)456 2044 y(sidering)40 b(prop)s(erties)h(of)e Fk(E)9 b(R)40 b Fs(as)g(a)f(whole,)j(some)f(examples)h(are)d (considered.)456 2160 y(The)30 b(sequences)i(that)c(arise)i(here)g(are) f(therefore)g(close)h(in)f(spirit)h(to)f(the)g(`eigen-)456 2281 y(sequences')37 b(for)c(the)i(transformation)f Fo(M)2033 2257 y(\177)2019 2281 y(OBIUS)g Fs(discussed)j(in)d([)p 0 1 0 0 TeXcolorcmyk(1)p Black (#cite.MR96i:05004) [[429 443 435 455] [1 1 1 [3 3]] [0 0 1]] pdfm (]) g(with)h(the)456 2398 y(additional)28 b(requiremen)m(t)i(that)e(the)h (sequence)i Fp(f)2279 2361 y Fq(\003)2346 2398 y Fs(b)s(e)d(divisible)i (b)m(y)f Fp(n)f Fs(and)h(non-)456 2514 y(negativ)m(e.)p Black 456 2705 a Fo(Example)38 b(2.2.)p Black Black 190 w Fs(\(1\))p Black 42 w(The)31 b(Fib)s(onacci)h(sequence)p 0 1 0 0 TeXcolorcmyk 33 w(A000045)p Black [[371 392 415 404] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000045) pdfm 30 w(is)f(not)g(in)g Fk(E)9 b(R)p Fs(.)770 2821 y(Using)43 b(\()p 0 1 0 0 TeXcolorcmyk(3)p Black (#equation.3) [[203 378 209 390] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(w)m (e)h(see)h(that)e Fp(f)1811 2836 y Fi(3)1879 2821 y Fk(\000)29 b Fp(f)2033 2836 y Fi(1)2115 2821 y Fs(m)m(ust)44 b(alw)m(a)m(ys)f(b)s (e)g(divisible)h(b)m(y)g(3,)770 2937 y(but)h(the)g(Fib)s(onacci)f (sequence)k(b)s(egins)d(1)p Fp(;)17 b Fs(1)p Fp(;)g Fs(2)p Fp(;)g Fs(3)p Fp(;)g(:)g(:)g(:)d Fs(.)79 b(By)45 b(con)m(trast)770 3054 y(the)37 b(golden)f(mean)h(shift)f(\(see)h([)p 0 1 0 0 TeXcolorcmyk(10)p Black (#cite.MR97a:58050) [[308 351 320 363] [1 1 1 [3 3]] [0 0 1]] pdfm (]\))f(sho)m(ws)i(that)e(the)g(closely)i(related)770 3170 y(Lucas)h(sequence)p 0 1 0 0 TeXcolorcmyk 42 w(A000204)p Black [[248 337 292 349] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000204) pdfm 37 w(is)g(in)g Fk(E)9 b(R)p Fs(.)61 b(This)40 b(will)f(b)s(e)g(dealt)g (with)g(in)770 3286 y(greater)33 b(generalit)m(y)h(in)e(Section)p 0 1 0 0 TeXcolorcmyk 34 w(2.2)p Black (#subsection.2.2) [[313 323 328 335] [1 1 1 [3 3]] [0 0 1]] pdfm 32 w(b)s(elo)m(w.)p Black 604 3402 a(\(2\))p Black 41 w(F)-8 b(or)32 b(an)m(y)h(map)g Fp(T)14 b Fs(,)32 b(equation)i(\()p 0 1 0 0 TeXcolorcmyk(3)p Black (#equation.3) [[302 309 307 321] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)e (when)i Fp(n)e Fs(is)h(a)g(prime)g Fp(p)p Fs(,)g(states)g(that)1485 3588 y Fp(f)1533 3603 y Ft(p)1573 3588 y Fs(\()p Fp(T)14 b Fs(\))27 b Fk(\021)i Fp(f)1901 3603 y Fi(1)1940 3588 y Fs(\()p Fp(T)14 b Fs(\))32 b(mo)s(d)h Fp(p:)770 3774 y Fs(If)48 b Fp(A)53 b Fk(2)h Fp(GL)1272 3789 y Ft(k)1315 3774 y Fs(\()p Fh(Z)p Fs(\))44 b(is)k(an)g(in)m(v)m(ertible)i(in)m (teger)e(matrix)g(with)g(no)g(unit)770 3890 y(ro)s(ot)32 b(eigen)m(v)-5 b(alues,)35 b(then)e(the)g(p)s(erio)s(dic)g(p)s(oin)m (ts)g(in)g(the)g(corresp)s(onding)770 4007 y(automorphism)h(of)e(the)h Fp(k)s Fs(-torus)f(sho)m(w)i(that)1256 4193 y(det)q(\()p Fp(A)1503 4152 y Ft(p)1565 4193 y Fk(\000)22 b Fp(I)8 b Fs(\))28 b Fk(\021)g Fs(det\()p Fp(A)22 b Fk(\000)h Fp(I)8 b Fs(\))32 b(mo)s(d)h Fp(p)770 4379 y Fs(for)f(all)h(primes)h Fp(p)p Fs(.)p Black 604 4495 a(\(3\))p Black 41 w(Similarly)-8 b(,)44 b(if)d Fp(B)46 b Fk(2)c Fp(M)1634 4510 y Ft(k)1677 4495 y Fs(\()p Fh(N)9 b Fs(\))47 b(is)41 b(a)g(matrix)g(of)f (non-negativ)m(e)i(in)m(tegers,)770 4611 y(the)33 b(asso)s(ciated)h (subshift)g(of)e(\014nite)h(t)m(yp)s(e)g(\(see)h([)p 0 1 0 0 TeXcolorcmyk(10)p Black (#cite.MR97a:58050) [[381 164 392 176] [1 1 1 [3 3]] [0 0 1]] pdfm (]\))e(sho)m(ws)i(that)1347 4797 y(trace\()p Fp(B)1675 4756 y Ft(p)1715 4797 y Fs(\))28 b Fk(\021)g Fs(trace\()p Fp(B)5 b Fs(\))33 b(mo)s(d)f Fp(p)770 4983 y Fs(for)39 b(all)g(primes)i Fp(p)p Fs(.)64 b(When)41 b Fp(k)h Fs(=)d(1)h(this)g (is)g(F)-8 b(ermat's)40 b(little)f(theorem.)770 5099 y(When)33 b Fp(B)g Fs(=)27 b([2],)32 b(so)g Fp(f)1593 5114 y Ft(n)1668 5099 y Fs(=)c(2)1821 5063 y Ft(n)1867 5099 y Fs(,)33 b Fp(f)1986 5063 y Ft(o)1975 5124 y(n)2055 5099 y Fs(is)g(the)f(sequence)p 0 1 0 0 TeXcolorcmyk 35 w(A001037)p Black [[399 105 443 117] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001037) pdfm 30 w(\(shifted)770 5216 y(b)m(y)i(one\))e(coun)m(ting)i(irreducible)g (p)s(olynomials)g(of)e(degree)h Fp(n)g Fs(o)m(v)m(er)h Fh(F)3314 5231 y Fi(2)3360 5216 y Fs(.)p Black Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black 456 251 a Fm(4)p Black Black 604 450 a Fs(\(4\))p Black 41 w(The)36 b(subshifts)h(of)e (\014nite)h(t)m(yp)s(e)g(giv)m(e)g(a)f(family)g(of)g(elemen)m(ts)i(of)e Fk(E)9 b(R)35 b Fs(of)770 566 y(exp)s(onen)m(tial)30 b(t)m(yp)s(e.)43 b(Another)29 b(family)h(comes)f(from)g(P)m(ascal's)h (triangle:)770 684 y(if)38 b Fp(k)j(>)c Fs(1,)i(1)e Fk(\024)h Fp(j)43 b(<)37 b(k)k Fs(and)e Fp(a)1921 699 y Ft(n)2005 684 y Fs(=)2118 604 y Fr(\000)2164 640 y Ft(k)r(n)2167 719 y(j)t(n)2245 604 y Fr(\001)2291 684 y Fs(,)h(then)e Fp(a)g Fk(2)g(E)9 b(R)o Fs(.)61 b(F)-8 b(or)37 b Fp(k)j Fs(=)e(2)770 815 y(and)i Fp(j)47 b Fs(=)40 b(1,)i(if)d Fp(f)1432 830 y Ft(n)1520 815 y Fs(=)h Fp(f)1684 830 y Ft(n)1731 815 y Fs(\()p Fp(T)14 b Fs(\))40 b(for)f(the)i(realizing)f (map)g Fp(T)14 b Fs(,)42 b(then)f Fp(f)3300 778 y Fq(\003)3289 839 y Ft(n)3379 815 y Fs(is)770 931 y(the)25 b(sequence)p 0 1 0 0 TeXcolorcmyk 28 w(A007727)p Black [[231 605 275 617] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=007727) pdfm 23 w(coun)m(ting)h(2)p Fp(n)p Fs(-b)s(ead)e(blac)m(k)i(and)f(white)h (strings)770 1047 y(with)33 b Fp(n)g Fs(blac)m(k)h(b)s(eads)f(and)g (fundamen)m(tal)g(p)s(erio)s(d)g(2)p Fp(n)p Fs(.)p Black 604 1163 a(\(5\))p Black 41 w(Connected)38 b Fp(S)6 b Fs(-in)m(teger)37 b(dynamical)g(systems)i(\(see)f([)p 0 1 0 0 TeXcolorcmyk(3)p Black (#cite.MR99b:11089) [[402 577 407 589] [1 1 1 [3 3]] [0 0 1]] pdfm (],)f([)p 0 1 0 0 TeXcolorcmyk(13)p Black (#cite.MR99k:58152) [[422 577 433 589] [1 1 1 [3 3]] [0 0 1]] pdfm (]) f(for)g(these)770 1280 y(and)h(the)f(next)i(example\):)52 b(a)36 b(subset)i Fp(S)i Fk(\032)35 b(f)p Fs(2)p Fp(;)17 b Fs(3)p Fp(;)g Fs(5)p Fp(;)g Fs(7)p Fp(;)g Fs(11)p Fp(;)g(:)g(:)g(:)c Fk(g)36 b Fs(and)h(a)770 1396 y(rational)h Fp(\030)j Fk(6)p Fs(=)36 b(0)i(are)g(giv)m(en)h(with)g(the)f(prop)s(ert)m(y)h (that)e Fk(j)p Fp(\030)5 b Fk(j)2975 1411 y Ft(p)3051 1396 y Fp(>)36 b Fs(1)74 b(=)-17 b Fk(\))770 1512 y Fp(p)31 b Fk(2)h Fp(S)6 b Fs(.)49 b(The)36 b(resulting)g(system)g(constructs)g (a)f(map)g Fp(T)44 b Fs(:)32 b Fp(X)39 b Fk(!)31 b Fp(X)42 b Fs(for)770 1628 y(whic)m(h)1481 1795 y Fp(f)1529 1810 y Ft(n)1576 1795 y Fs(\()p Fp(T)14 b Fs(\))28 b(=)1871 1700 y Fr(Y)1854 1911 y Ft(p)p Fq(\0241)2032 1795 y Fk(j)p Fp(\030)2108 1754 y Ft(n)2176 1795 y Fk(\000)23 b Fs(1)p Fk(j)2353 1810 y Ft(p)2392 1795 y Fp(:)770 2088 y Fs(With)33 b Fp(\030)f Fs(=)c(2)p Fp(;)17 b(S)33 b Fs(=)27 b Fk(f)p Fs(2)p Fp(;)17 b Fs(3)p Fp(;)g Fs(5)p Fp(;)g Fs(7)p Fk(g)31 b Fs(this)i(giv)m(es)h(the)f(sequence)703 2295 y(1)p Fp(;)17 b Fs(1)p Fp(;)g Fs(1)p Fp(;)g Fs(1)p Fp(;)g Fs(31)p Fp(;)g Fs(1)p Fp(;)g Fs(127)p Fp(;)g Fs(17)p Fp(;)g Fs(73)p Fp(;)g Fs(341)p Fp(;)f Fs(2047)o Fp(;)h Fs(13)p Fp(;)f Fs(8191)o Fp(;)h Fs(546)o(1)p Fp(;)g Fs(468)o(1)p Fp(;)g(:)f(:)h(:)770 2503 y Fs(in)33 b Fk(E)9 b(R)p Fs(.)p Black 604 2619 a(\(6\))p Black 41 w(Zero-dimensional)49 b Fp(S)6 b Fs(-in)m(teger)47 b(dynamical)i(systems:)76 b(a)48 b(prime)g Fp(p)g Fs(is)770 2735 y(\014xed,)29 b(a)d(subset)i Fp(S)k Fs(of)26 b(the)h(set)g(of)e (all)i(irreducible)h(p)s(olynomials)f(in)f Fh(F)3309 2750 y Ft(p)3355 2735 y Fs([)p Fp(t)p Fs(])770 2851 y(and)36 b(a)g(rational)g(function)g Fp(\030)i Fk(2)c Fh(F)2036 2866 y Ft(p)2082 2851 y Fs(\()p Fp(t)p Fs(\))i(are)g(giv)m(en,)i(with)f (the)g(prop)s(ert)m(y)770 2968 y(that)g Fk(j)p Fp(\030)5 b Fk(j)1090 2983 y Ft(f)1171 2968 y Fp(>)36 b Fs(1)72 b(=)-17 b Fk(\))72 b Fp(f)46 b Fk(2)37 b Fp(S)6 b Fs(.)58 b(The)39 b(resulting)f(system)i(constructs)f(a)770 3084 y(map)33 b Fp(T)41 b Fs(:)28 b Fp(X)36 b Fk(!)27 b Fp(X)40 b Fs(for)32 b(whic)m(h)1228 3307 y Fp(f)1276 3322 y Ft(n)1323 3307 y Fs(\()p Fp(T)14 b Fs(\))27 b(=)h Fk(j)p Fp(\030)1677 3266 y Ft(n)1745 3307 y Fk(\000)23 b Fs(1)p Fk(j)1922 3324 y Ft(t)1947 3305 y Fg(\000)p Ff(1)2055 3307 y Fk(\002)2159 3213 y Fr(Y)2155 3425 y Ft(f)7 b Fq(2)p Ft(S)2306 3307 y Fk(j)p Fp(\030)2382 3266 y Ft(n)2451 3307 y Fk(\000)22 b Fs(1)p Fk(j)2627 3322 y Ft(f)770 3622 y Fs(where)29 b Fk(j)12 b(\001)g(j)1155 3639 y Ft(t)1180 3620 y Fg(\000)p Ff(1)1295 3622 y Fs(is)28 b(used)h(to)f(denote)g(the)g(v)-5 b(aluation)28 b(`at)f(in\014nit)m(y')j(induced)770 3738 y(b)m(y)39 b Fk(j)p Fp(t)p Fk(j)1002 3755 y Ft(t)1027 3736 y Fg(\000)p Ff(1)1150 3738 y Fs(=)d Fp(p)p Fs(.)58 b(T)-8 b(aking)38 b Fp(p)e Fs(=)g(2,)j Fp(S)j Fs(=)36 b Fk(f)p Fp(t)26 b Fk(\000)g Fs(1)p Fk(g)37 b Fs(and)h Fp(\030)i Fs(=)c Fp(t)i Fs(giv)m(es)h(the)770 3855 y(form)m(ula)1566 4021 y Fp(f)1614 4036 y Ft(n)1661 4021 y Fs(\()p Fp(T)14 b Fs(\))28 b(=)f(2)1988 3980 y Ft(n)p Fq(\000)p Fi(2)2121 3957 y Ff(ord)2209 3972 y(2)2244 3957 y(\(n\))770 4209 y Fs(and)33 b(the)g(sequence)p 0 1 0 0 TeXcolorcmyk 35 w(A059991)p Black [[256 212 300 224] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=059991) pdfm 31 w(in)g Fk(E)9 b(R)p Fs(.)456 4412 y(2.1.)48 b Fo(Algebra)39 b(in)g Fk(E)9 b(R)p Fo(.)49 b Fs(The)34 b(set)h Fk(E)9 b(R)33 b Fs({)h(or)f(the)h(ring)f Fp(K)2628 4427 y Fi(0)2668 4412 y Fs(\()p Fk(E)9 b(R)p Fs(\))33 b({)h(has)g(a)f(v)m(ery)456 4528 y(ric)m(h)40 b(structure.)64 b(Sa)m(y)40 b(that)f(a)g(sequence)i Fp(a)f Fk(2)f(E)9 b(R)39 b Fu(factorizes)h Fs(if)f(there)h(exists)456 4644 y(sequences)d Fp(b;)17 b(c)30 b Fk(2)h(E)9 b(R)34 b Fs(with)h Fp(a)1609 4659 y Ft(n)1686 4644 y Fs(=)30 b Fp(b)1833 4659 y Ft(n)1880 4644 y Fp(c)1922 4659 y Ft(n)2003 4644 y Fs(for)k(all)g Fp(n)c Fk(\025)h Fs(1,)j(and)g(is)h Fu(prime)g Fs(if)f(suc)m(h)456 4760 y(a)e(factorization)g(requires)j (one)d(of)h Fp(b)g Fs(or)f Fp(c)g Fs(to)g(b)s(e)h(the)g(constan)m(t)h (sequence)h(\(1\).)p Black 456 4964 a Fo(Lemma)k(2.3.)p Black 42 w Fk(E)9 b(R)35 b Fv(c)-5 b(ontains)34 b(the)h(c)-5 b(onstant)35 b(se)-5 b(quenc)g(es)34 b(and)h(is)f(close)-5 b(d)35 b(under)456 5080 y(addition)30 b(and)h(multiplic)-5 b(ation.)43 b(Elements)30 b(of)h Fk(E)9 b(R)31 b Fv(may)g(have)g (in\014nitely)g(many)456 5196 y(non-trivial)j(factors.)44 b(Ther)-5 b(e)34 b(ar)-5 b(e)35 b(non-trivial)f(primes)g(in)g Fk(E)9 b(R)p Fv(.)p Black Black eop %%Page: 5 5 5 4 bop Black 0 TeXcolorgray Black 3406 251 a Fm(5)p Black 456 450 a Fs(Pro)s(of.)32 b(The)d(constan)m(t)h(sequence)h(\(1\)) d(is)i(in)e Fk(E)9 b(R)29 b Fs(since)h(it)f(is)g(realized)h(b)m(y)f (taking)456 566 y Fp(X)55 b Fs(to)47 b(b)s(e)g(a)g(singleton.)89 b(The)48 b(condition)g(in)f(Lemma)p 0 1 0 0 TeXcolorcmyk 48 w(2.1)p Black (#theorem.2.1) [[386 649 401 661] [1 1 1 [3 3]] [0 0 1]] pdfm 47 w(is)h(closed)h(under)456 683 y(addition.)41 b(On)26 b(the)g(other)h(hand,)g(if)f Fp(\036)g Fs(and)g Fp( )k Fs(are)c(exactly)h(realized)g(b)m(y)g(systems)456 799 y(\()p Fp(X)r(;)17 b(T)d Fs(\))34 b(and)g(\()p Fp(Y)5 b(;)17 b(S)6 b Fs(\),)35 b(then)g(\()p Fp(X)c Fk(\002)24 b Fp(Y)5 b(;)17 b(T)37 b Fk(\002)24 b Fp(S)6 b Fs(\))34 b(exactly)i(realizes)g(\()p Fp(\036)2957 814 y Ft(n)3027 799 y Fk(\001)23 b Fp( )3141 814 y Ft(n)3188 799 y Fs(\).)49 b(F)-8 b(or)456 933 y(eac)m(h)28 b Fp(k)j Fk(\025)d Fs(1)f(de\014ne)i (a)e(sequence)j Fp(r)1732 897 y Fi(\()p Ft(k)r Fi(\))1857 933 y Fs(b)m(y)e Fp(r)2034 882 y Fi(\()p Ft(k)r Fi(\))2031 943 y Ft(n)2159 933 y Fs(=)g(0)f(for)g(1)g Fp(<)h(n)g Fk(\024)g Fp(k)j Fs(and)c Fp(r)3167 882 y Fi(\()p Ft(k)r Fi(\))3164 943 y Ft(n)3292 933 y Fs(=)h(1)456 1061 y(for)37 b Fp(n)f(>)g(k)41 b Fs(or)c Fp(n)g Fs(=)f(1.)58 b(Then)39 b Fp(a)1684 1025 y Fi(\()p Ft(k)r Fi(\))1818 1061 y Fk(2)e(E)9 b(R)p Fs(,)39 b(where)g Fp(a)2471 1010 y Fi(\()p Ft(k)r Fi(\))2471 1071 y Ft(n)2605 1061 y Fs(=)2717 986 y Fr(P)2822 1090 y Ft(d)p Fq(j)p Ft(n)2942 1061 y Fp(dr)3040 1010 y Fi(\()p Ft(k)r Fi(\))3037 1089 y Ft(d)3137 1061 y Fs(.)58 b(Since)456 1216 y(for)32 b(eac)m(h)i Fp(n)g Fs(the)f(sequence)j Fp(a)1541 1165 y Fi(\(1\))1541 1226 y Ft(n)1635 1216 y Fp(;)17 b(a)1730 1165 y Fi(\(2\))1730 1226 y Ft(n)1825 1216 y Fp(;)g(a)1920 1165 y Fi(\(3\))1920 1226 y Ft(n)2014 1216 y Fp(;)g(:)g(:)g(:)49 b Fs(has)33 b(only)h(\014nitely)g(man)m(y)g (terms)456 1332 y(not)g(equal)i(to)f(1,)g(the)h(pro)s(duct)1663 1257 y Fr(Q)1757 1361 y Ft(k)r Fq(\025)p Fi(1)1907 1332 y Fp(a)1958 1296 y Fi(\()p Ft(k)r Fi(\))2087 1332 y Fs(=)c(\(1)p Fp(;)17 b Fs(3)p Fp(;)g Fs(16)p Fp(;)g Fs(245)p Fp(;)g Fs(1296)p Fp(;)g Fs(41160)p Fp(;)g(:)g(:)g(:)10 b Fs(\))456 1450 y(is)37 b(an)g(elemen)m(t)i(of)e Fk(E)9 b(R)37 b Fs(with)h(in\014nitely)g(man)m(y)h(non-trivial)e(factors.)57 b(Finally)-8 b(,)456 1566 y(the)33 b(sequence)i(\(1)p Fp(;)17 b Fs(3)p Fp(;)g Fs(1)p Fp(;)g Fs(3)p Fp(;)g(:)g(:)g(:)d Fs(\))32 b(is)h(a)g(non-trivial)f(prime)i(in)e Fk(E)9 b(R)p Fs(.)511 b Fe(2)555 1682 y Fs(In)39 b([)p 0 1 0 0 TeXcolorcmyk(9)p Black (#cite.MR86c:58092) [[157 515 163 527] [1 1 1 [3 3]] [0 0 1]] pdfm (,) g(Sect.)61 b(6])39 b(a)e(p)s(erio)s(dic)i(p)s(oin)m(t)f(coun)m(ting)h (argumen)m(t)g(is)g(used)g(to)f(sho)m(w)456 1799 y(that)28 b(the)h(full)g Fp(p)p Fs(-shift,)h(for)e Fp(p)h Fs(a)f(prime,)j(is)e (not)g(top)s(ologically)g(conjugate)f(to)h(the)456 1915 y(direct)42 b(pro)s(duct)g(of)f(t)m(w)m(o)h(dynamical)h(systems.)73 b(In)41 b(that)h(argumen)m(t,)i(sp)s(ecial)456 2031 y(prop)s(erties)c (of)e(subshifts)j(of)e(\014nite)h(t)m(yp)s(e)g(are)f(needed)i(\(sp)s (eci\014cally)-8 b(,)43 b(the)d(fact)456 2147 y(that)h Fp(f)724 2166 y Ft(p)760 2147 y Fd(k)9 b Fs(\()p Fp(T)14 b Fs(\))43 b(=)g(1)f(for)f(all)h Fp(k)47 b Fk(\025)c Fs(1)f(implies)h(that)f Fp(f)2423 2162 y Ft(n)2470 2147 y Fs(\()p Fp(T)14 b Fs(\))43 b(=)g(1)f(for)f(all)h Fp(n)i Fk(\025)g Fs(1)456 2263 y(for)i(suc)m(h)j(systems\).)89 b(This)49 b(result)f(do)s(es)g(not)f(follo)m(w)g(from)g(the)g (arithmetic)456 2380 y(of)37 b Fk(E)9 b(R)38 b Fs(alone:)54 b(for)37 b(example,)k(\(3)1716 2344 y Ft(n)1763 2380 y Fs(\))c Fk(2)g(E)9 b(R)37 b Fs(factorizes)i(in)m(to)f(\(1)p Fp(;)17 b Fs(3)p Fp(;)g Fs(1)p Fp(;)g Fs(3)p Fp(;)g(:)g(:)g(:)d Fs(\))26 b Fk(\002)456 2496 y Fs(\(3)p Fp(;)17 b Fs(3)p Fp(;)g Fs(3)729 2460 y Fi(3)767 2496 y Fp(;)g Fs(3)860 2460 y Fi(3)899 2496 y Fp(;)g(:)g(:)g(:)e Fs(\))25 b(in)f Fk(E)9 b(R)24 b Fs(\(neither)i(of)e(whic)m(h)h(can)g(b)s(e)g(realized)g (using)g(a)f(subshift)456 2612 y(of)31 b(\014nite)h(t)m(yp)s(e\).)44 b(A)32 b(similar)g(factorization)f(is)h(p)s(ossible)h(for)e(\()p Fp(p)2796 2576 y Ft(n)2843 2612 y Fs(\))h(and)f(an)m(y)i(o)s(dd)456 2728 y(prime)g Fp(p)f Fs(\(see)i([)p 0 1 0 0 TeXcolorcmyk(11)p Black (#cite.puri-thesis) [[196 390 208 402] [1 1 1 [3 3]] [0 0 1]] pdfm (]) e(for)g(the)h(details\).)p Black 456 3004 a Fo(Lemma)c(2.4.)p Black 35 w Fv(Ther)-5 b(e)27 b(ar)-5 b(e)26 b(no)h(non-c)-5 b(onstant)26 b(p)-5 b(olynomials)26 b(in)g Fk(E)9 b(R)p Fv(.)42 b(Ther)-5 b(e)27 b(ar)-5 b(e)456 3120 y(non-trivial)32 b(multiplic)-5 b(ative)33 b(se)-5 b(quenc)g(es)33 b(in)h Fk(E)9 b(R)p Fv(,)33 b(but)i(ther)-5 b(e)33 b(ar)-5 b(e)34 b(no)f(c)-5 b(ompletely)456 3236 y(multiplic)g(ative)34 b(se)-5 b(quenc)g(es)34 b(ap)-5 b(art)35 b(fr)-5 b(om)34 b(the)h(c)-5 b(onstant)34 b(se)-5 b(quenc)g(e)34 b Fs(\(1\))p Fv(.)456 3512 y Fs(Pro)s(of.)e(Assume)i(that)1337 3845 y Fp(P)14 b Fs(\()p Fp(n)p Fs(\))28 b(=)f Fp(c)1721 3860 y Fi(0)1783 3845 y Fs(+)22 b Fp(c)1923 3860 y Fi(1)1962 3845 y Fp(n)h Fs(+)f Fk(\001)17 b(\001)g(\001)j Fs(+)i Fp(c)2419 3860 y Ft(k)2462 3845 y Fp(n)2520 3804 y Ft(k)456 4179 y Fs(with)43 b Fp(c)730 4194 y Ft(k)818 4179 y Fk(6)p Fs(=)i(0,)g Fp(k)j Fk(\025)d Fs(1,)g(and)e(that)g(\()p Fp(P)14 b Fs(\()p Fp(n)p Fs(\)\))44 b Fk(2)i(E)9 b(R)p Fs(.)74 b(After)42 b(m)m(ultiplying)j(the)456 4295 y(divisibilit)m(y)35 b(condition)e(\()p 0 1 0 0 TeXcolorcmyk(3)p Black (#equation.3) [[240 202 246 214] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(b)m (y)h(the)g(least)g(common)g(m)m(ultiple)h(of)e(the)h(denomi-)456 4411 y(nators)27 b(of)f(the)i(\(rational\))f(co)s(e\016cien)m(ts)i(of)e Fp(P)14 b Fs(,)27 b(w)m(e)h(pro)s(duce)g(a)f(p)s(olynomial)h(with)456 4528 y(in)m(teger)h(co)s(e\016cien)m(ts)h(satisfying)f(\()p 0 1 0 0 TeXcolorcmyk(3)p Black (#equation.3) [[280 174 286 186] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)42 b(It)28 b(is)h(therefore)f(enough)h(to)f(assume)i(that)456 4644 y(the)41 b(co)s(e\016cien)m(ts)h Fp(c)1175 4659 y Ft(i)1244 4644 y Fs(are)e(all)h(in)m(tegers.)68 b(Let)41 b(\()p Fp(f)2249 4659 y Ft(n)2296 4644 y Fs(\))f(and)h(\()p Fp(f)2669 4608 y Fq(\003)2658 4669 y Ft(n)2708 4644 y Fs(\))f(b)s(e)h(the)g(p)s(erio)s(dic)456 4760 y(p)s(oin)m(ts)g(and)h (least)g(p)s(erio)s(dic)f(p)s(oin)m(ts)h(in)g(the)f(corresp)s(onding)h (system)i(\()p Fp(X)r(;)17 b(T)d Fs(\),)456 4876 y(and)32 b(let)h Fp(p)g Fs(b)s(e)f(an)m(y)i(prime.)44 b(By)33 b(\()p 0 1 0 0 TeXcolorcmyk(2)p Black (#equation.2) [[275 132 281 144] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)1644 5210 y Fp(f)1703 5169 y Fq(\003)1692 5236 y Ft(p)1728 5217 y Ff(2)1794 5210 y Fs(=)27 b Fp(f)1945 5227 y Ft(p)1981 5208 y Ff(2)2042 5210 y Fk(\000)22 b Fp(f)2189 5225 y Ft(p)2229 5210 y Fp(;)p Black Black eop %%Page: 6 6 6 5 bop Black 0 TeXcolorgray Black 456 251 a Fm(6)p Black 456 450 a Fs(so)456 651 y Fp(f)515 610 y Ft(o)504 677 y(p)540 658 y Ff(2)606 651 y Fs(=)719 569 y Fp(f)778 532 y Fq(\003)767 599 y Ft(p)803 580 y Ff(2)p 719 628 123 4 v 736 719 a Fp(p)785 691 y Fi(2)934 651 y Fs(=)1103 583 y Fp(f)1151 599 y Ft(p)1187 580 y Ff(2)1248 583 y Fk(\000)22 b Fp(f)1395 598 y Ft(p)p 1103 628 332 4 v 1225 719 a Fp(p)1274 691 y Fi(2)934 908 y Fs(=)1123 840 y(1)p 1103 885 89 4 v 1103 976 a Fp(p)1152 947 y Fi(2)1218 827 y Fr(\000)1264 908 y Fp(c)1306 923 y Fi(1)1345 908 y Fp(p)1394 867 y Fi(2)1456 908 y Fs(+)g Fp(c)1596 923 y Fi(2)1635 908 y Fp(p)1684 867 y Fi(4)1746 908 y Fs(+)g Fk(\001)17 b(\001)g(\001)j Fs(+)i Fp(c)2122 923 y Ft(k)2165 908 y Fp(p)2214 867 y Fi(2)p Ft(k)2314 908 y Fk(\000)h Fs(\()p Fp(c)2494 923 y Fi(1)2533 908 y Fp(p)f Fs(+)g Fp(c)2744 923 y Fi(2)2783 908 y Fp(p)2832 867 y Fi(2)2894 908 y Fs(+)g Fk(\001)17 b(\001)g(\001)j Fs(+)i Fp(c)3270 923 y Ft(k)3313 908 y Fp(p)3362 867 y Ft(k)3405 908 y Fs(\))3443 827 y Fr(\001)939 1143 y Fk(2)88 b(\000)1180 1076 y Fp(c)1222 1091 y Fi(1)p 1180 1120 82 4 v 1197 1212 a Fp(p)1294 1143 y Fs(+)22 b Fh(Z)p Fp(;)456 1349 y Fs(and)32 b(therefore)h Fp(p)g Fs(divides)h Fp(c)1506 1364 y Fi(1)1578 1349 y Fs(for)e(all)h(primes)h Fp(p)p Fs(,)e(sho)m(wing)i(that)e Fp(c)2911 1364 y Fi(1)2978 1349 y Fs(=)c(0.)555 1466 y(No)m(w)33 b(let)g Fp(q)k Fs(b)s(e)32 b(another)h(prime,)h(and)e(recall)h(that)574 1623 y Fp(\026)p Fs(\(1\))27 b(=)h(1)p Fp(;)17 b(\026)p Fs(\()p Fp(p)p Fs(\))26 b(=)i Fk(\000)p Fs(1)p Fp(;)17 b(\026)p Fs(\()p Fp(q)t Fs(\))27 b(=)g Fk(\000)p Fs(1)p Fp(;)17 b(\026)p Fs(\()p Fp(p)2094 1582 y Fi(2)2133 1623 y Fs(\))28 b(=)g(0)p Fp(;)17 b(\026)p Fs(\()p Fp(p)2542 1582 y Fi(2)2580 1623 y Fp(q)t Fs(\))27 b(=)h(0)p Fp(;)17 b(\026)p Fs(\()p Fp(pq)t Fs(\))27 b(=)g(1)p Fp(:)456 1780 y Fs(Since)33 b Fp(c)752 1795 y Fi(1)819 1780 y Fs(=)28 b(0,)1149 1937 y Fp(f)1197 1952 y Ft(n)1272 1937 y Fs(=)f Fp(c)1417 1952 y Fi(0)1479 1937 y Fs(+)22 b Fp(n)1635 1896 y Fi(2)1675 1937 y Fs(\()p Fp(c)1755 1952 y Fi(2)1816 1937 y Fs(+)g Fp(c)1956 1952 y Fi(3)1995 1937 y Fp(n)h Fs(+)f Fk(\001)17 b(\001)g(\001)j Fs(+)i Fp(c)2452 1952 y Ft(k)2495 1937 y Fp(n)2553 1896 y Ft(k)r Fq(\000)p Fi(2)2686 1937 y Fs(\))p Fp(;)569 b Fs(\(4\))456 2094 y(and)32 b(b)m(y)i(\()p 0 1 0 0 TeXcolorcmyk(3)p Black (#equation.3) [[170 466 176 478] [1 1 1 [3 3]] [0 0 1]] pdfm (\))1283 2213 y Fp(p)1332 2171 y Fi(2)1371 2213 y Fp(q)1418 2128 y Fr(\014)1418 2188 y(\014)1451 2213 y Fp(f)1499 2229 y Ft(p)1535 2210 y Ff(2)1569 2229 y Ft(q)1629 2213 y Fk(\000)23 b Fp(f)1777 2228 y Ft(pq)1873 2213 y Fk(\000)g Fp(f)2021 2229 y Ft(p)2057 2210 y Ff(2)2117 2213 y Fs(+)f Fp(f)2263 2228 y Ft(p)2330 2213 y Fs(=)28 b Fp(f)2493 2171 y Fq(\003)2482 2238 y Ft(p)2518 2219 y Ff(2)2552 2238 y Ft(q)2590 2213 y Fp(;)456 2349 y Fs(so)1690 2427 y Fp(f)1738 2444 y Ft(p)1774 2425 y Ff(2)1834 2427 y Fk(\000)23 b Fp(f)1982 2442 y Ft(p)p 1690 2473 332 4 v 1788 2564 a Fp(p)1837 2535 y Fi(2)1877 2564 y Fp(q)2059 2496 y Fk(2)29 b Fh(Z)456 2687 y Fs(b)m(y)k(\()p 0 1 0 0 TeXcolorcmyk(4)p Black (#equation.4) [[147 395 153 407] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)43 b(It)33 b(follo)m(ws)g(that)640 2848 y Fp(c)682 2863 y Fi(0)721 2848 y Fs(\(1)22 b Fk(\000)g Fs(1\))g(+)g Fp(c)1178 2863 y Fi(2)1218 2848 y Fs(\()p Fp(p)1305 2807 y Fi(4)1366 2848 y Fk(\000)h Fp(p)1515 2807 y Fi(2)1554 2848 y Fs(\))f(+)g Fp(c)1754 2863 y Fi(3)1794 2848 y Fs(\()p Fp(p)1881 2807 y Fi(6)1942 2848 y Fk(\000)h Fp(p)2091 2807 y Fi(3)2130 2848 y Fs(\))f(+)g Fk(\001)17 b(\001)g(\001)k Fs(+)h Fp(c)2567 2863 y Ft(k)2609 2848 y Fs(\()p Fp(p)2696 2807 y Fi(2)p Ft(k)2796 2848 y Fk(\000)h Fp(p)2945 2807 y Ft(k)2988 2848 y Fs(\))k Fk(2)h Fp(q)t Fh(Z)456 3006 y Fs(for)k(all)g(primes)i Fp(q)i Fs(and)d Fp(p)f Fs(\(since)i Fp(f)1730 3022 y Ft(p)1766 3003 y Ff(2)1827 3006 y Fk(\000)23 b Fp(f)1975 3021 y Ft(p)2047 3006 y Fs(is)33 b(certainly)h(divisible)g (b)m(y)g Fp(p)3116 2969 y Fi(2)3155 3006 y Fs(\).)44 b(So)653 3172 y Fp(c)695 3187 y Fi(0)735 3172 y Fs(\(1)22 b Fk(\000)g Fs(1\))g(+)g Fp(c)1192 3187 y Fi(2)1232 3172 y Fs(\()p Fp(p)1319 3131 y Fi(4)1380 3172 y Fk(\000)h Fp(p)1529 3131 y Fi(2)1568 3172 y Fs(\))f(+)g Fp(c)1768 3187 y Fi(3)1808 3172 y Fs(\()p Fp(p)1895 3131 y Fi(6)1956 3172 y Fk(\000)h Fp(p)2105 3131 y Fi(3)2144 3172 y Fs(\))f(+)g Fk(\001)17 b(\001)g(\001)k Fs(+)h Fp(c)2581 3187 y Ft(k)2623 3172 y Fs(\()p Fp(p)2710 3131 y Fi(2)p Ft(k)2810 3172 y Fk(\000)h Fp(p)2959 3131 y Ft(k)3001 3172 y Fs(\))28 b(=)g(0;)456 3329 y(taking)39 b(the)h(limit)g(as)g Fp(p)f Fk(!)g(1)g Fs(of)1833 3290 y Fi(1)p 1799 3306 105 4 v 1799 3364 a Ft(p)1835 3345 y Ff(2)p Fd(k)1913 3329 y Fs(\()p Fp(f)1999 3346 y Ft(p)2035 3327 y Ff(2)2100 3329 y Fk(\000)28 b Fp(f)2253 3344 y Ft(p)2292 3329 y Fs(\))40 b(sho)m(ws)h(that)e Fp(c)2917 3344 y Ft(k)2999 3329 y Fs(=)g(0.)64 b(This)456 3454 y(con)m(tradiction)33 b(pro)m(v)m(es)h (the)f(\014rst)g(statemen)m(t.)555 3570 y(There)42 b(are)f(man)m(y)g(m) m(ultiplicativ)m(e)i(sequences)h(in)d Fk(E)9 b(R)p Fs(:)59 b(if)41 b Fp(f)2874 3534 y Fq(\003)2954 3570 y Fs(is)g(an)m(y)g(m)m (ul-)456 3686 y(tiplicativ)m(e)g(sequence,)j(then)d(so)f(is)g(the)g (corresp)s(onding)h(sequence)h Fp(f)51 b Fs(\(see)40 b([)p 0 1 0 0 TeXcolorcmyk(7)p Black (#cite.MR81i:10002) [[476 275 482 287] [1 1 1 [3 3]] [0 0 1]] pdfm (,) 456 3803 y(Theorem)d(265]\).)53 b(A)36 b(m)m(ultiplicativ)m(e)i (sequence)h Fp(\036)c Fs(is)i Fu(completely)g(m)m(ultiplica-)456 3919 y(tiv)m(e)27 b Fs(if)f Fp(\036)780 3934 y Ft(nm)916 3919 y Fs(=)i Fp(\036)1078 3934 y Ft(n)1125 3919 y Fp(\036)1183 3934 y Ft(m)1275 3919 y Fs(for)e(all)g Fp(n;)17 b(m)27 b Fk(\025)i Fs(1.)41 b(Assume)28 b(that)d Fp(\036)j Fk(2)g(E)9 b(R)26 b Fs(is)g(completely)456 4035 y(m)m(ultiplicativ)m(e,)43 b(with)c Fp(f)49 b Fs(the)40 b(realising)f(sequence.)65 b(F)-8 b(or)38 b Fp(p)h Fs(a)f(prime)i(and)f(an)m(y)456 4151 y Fp(r)30 b Fk(\025)e Fs(1,)1332 4268 y Fp(f)1391 4227 y Fq(\003)1380 4292 y Ft(p)1416 4274 y Fd(r)1483 4268 y Fs(=)f Fp(f)1634 4283 y Ft(p)1670 4264 y Fd(r)1731 4268 y Fk(\000)22 b Fp(f)1878 4285 y Ft(p)1914 4266 y Fd(r)r Fg(\000)p Ff(1)2059 4268 y Fs(=)27 b Fp(f)2221 4227 y Ft(r)2210 4292 y(p)2281 4268 y Fk(\000)c Fp(f)2440 4227 y Ft(r)r Fq(\000)p Fi(1)2429 4292 y Ft(p)456 4409 y Fs(b)m(y)33 b(\()p 0 1 0 0 TeXcolorcmyk(2)p Black (#equation.2) [[147 188 153 200] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)43 b(It)33 b(follo)m(ws)g(that)1616 4527 y Fp(p)1665 4486 y Ft(r)1703 4443 y Fr(\014)1703 4502 y(\014)1736 4527 y Fp(f)1795 4486 y Ft(r)r Fq(\000)p Fi(1)1784 4552 y Ft(p)1923 4527 y Fs(\()p Fp(f)2009 4542 y Ft(p)2071 4527 y Fk(\000)22 b Fs(1\))p Fp(:)456 4664 y Fs(With)33 b Fp(r)d Fs(=)d(1)33 b(this)g(implies)h(that)e Fp(f)1747 4679 y Ft(p)1815 4664 y Fs(=)27 b(1)22 b(+)g Fp(pk)2187 4679 y Ft(p)2259 4664 y Fs(for)32 b(all)h Fp(p)p Fs(,)f Fp(k)2703 4679 y Ft(p)2770 4664 y Fk(2)d Fh(N)2930 4679 y Fi(0)2976 4664 y Fs(.)43 b(No)m(w)1564 4826 y Fp(p)1613 4785 y Ft(r)1651 4741 y Fr(\014)1651 4801 y(\014)1684 4826 y Fs(\(1)22 b(+)g Fp(pk)1991 4841 y Ft(p)2030 4826 y Fs(\))2068 4785 y Ft(r)r Fq(\000)p Fi(1)2197 4826 y Fp(pk)2297 4841 y Ft(p)456 4983 y Fs(for)39 b(all)h Fp(p)g Fs(and)g Fp(r)i Fk(\025)f Fs(1.)65 b(It)40 b(follo)m(ws)h(that)f Fp(k)2097 4998 y Ft(p)2176 4983 y Fk(\021)h Fs(0)e(mo)s(d)h Fp(p)2658 4947 y Ft(r)2736 4983 y Fs(for)f(all)h Fp(r)j Fk(\025)e Fs(1,)g(so)456 5099 y Fp(k)507 5114 y Ft(p)580 5099 y Fs(=)33 b(0)j(for)f(all)h Fp(p)p Fs(.)54 b(It)36 b(follo)m(ws)h(that)f Fp(f)1891 5114 y Ft(p)1965 5099 y Fs(=)d(1)j(for)f(all)h(primes)i Fp(p)p Fs(,)f(so)f Fp(f)3053 5114 y Ft(n)3134 5099 y Fs(=)d(1)j(for)456 5216 y(all)c Fp(n)c Fk(\025)g Fs(1.)2512 b Fe(2)p Black Black eop %%Page: 7 7 7 6 bop Black 0 TeXcolorgray Black 3406 251 a Fm(7)p Black 555 450 a Fs(Examples)28 b(sho)m(w)e(that)g(the)f(additiv)m(e)i (con)m(v)m(olution)g(\()2512 375 y Fr(P)2617 479 y Ft(i)p Fi(+)p Ft(j)t Fi(=)p Ft(n)p Fi(+1)p Ft(;)p Fi(1)p Fq(\024)p Ft(i;j)t Fq(\024)p Ft(n)3221 450 y Fp(\036)3279 465 y Ft(i)3307 450 y Fp( )3370 465 y Ft(j)3406 450 y Fs(\))456 570 y(of)35 b(sequences)k Fp(\036;)17 b( )37 b Fk(2)d(E)9 b(R)36 b Fs(is)g(not)g(in)g(general)g(in)g Fk(E)9 b(R)p Fs(.)54 b(Similarly)-8 b(,)38 b(the)e(m)m(ulti-)456 686 y(plicativ)m(e)j(con)m(v)m(olution)g(\()1417 611 y Fr(P)1522 715 y Ft(d)p Fq(j)p Ft(n)1642 686 y Fp(\036)1700 701 y Ft(d)1740 686 y Fp( )1803 701 y Ft(n=d)1922 686 y Fs(\))f(is)g(not)g (in)g(general)g(in)g Fk(E)9 b(R)p Fs(.)60 b(There)39 b(is)456 814 y(also)f(no)g(closure)i(under)f(quotien)m(ts:)57 b(\(2)1972 778 y Ft(n)2019 814 y Fs(\))38 b Fk(2)g(E)9 b(R)38 b Fs(is)h(term-b)m(y-term)g(divisible)456 931 y(b)m(y)33 b(the)g(constan)m(t)h(sequence)h(\(2\))27 b Fk(2)h(E)9 b(R)p Fs(,)33 b(but)g(\(2)2275 895 y Ft(n)p Fq(\000)p Fi(1)2411 931 y Fs(\))40 b Fp(=)-61 b Fk(2)28 b(E)9 b(R)p Fs(.)456 1125 y(2.2.)48 b Fo(Binary)27 b(recurrence)f (sequences.)50 b Fs(In)24 b(this)f(section)h(w)m(e)g(expand)g(on)f(the) 456 1241 y(observ)-5 b(ation)32 b(made)h(in)g(Example)p 0 1 0 0 TeXcolorcmyk 34 w(2.2)p Black (#theorem.2.2) [[282 568 297 580] [1 1 1 [3 3]] [0 0 1]] pdfm (.1)e(b) m(y)j(sho)m(wing)f(that)f Fk(E)9 b(R)32 b Fs(only)h(con)m(tains)456 1357 y(sp)s(ecial)g(binary)g(recurrences.)p Black 456 1537 a Fo(Theorem)43 b(2.5.)p Black 44 w Fv(If)c Fs(\001)d(=)f Fp(a)1540 1501 y Fi(2)1605 1537 y Fs(+)25 b(4)p Fp(b)39 b Fv(is)g(not)g(a)g(squar)-5 b(e,)40 b(and)f Fs(\()p Fp(a;)17 b(a)2925 1501 y Fi(2)2989 1537 y Fs(+)26 b(2)p Fp(b)p Fs(\))35 b(=)h(1)p Fv(,)456 1654 y(then)e(a)h(se)-5 b(quenc)g(e)34 b Fp(u)g Fv(with)h Fp(u)1511 1669 y Fi(1)1550 1654 y Fp(;)17 b(u)1650 1669 y Fi(2)1716 1654 y Fk(\025)29 b Fs(1)34 b Fv(satisfying)h(the)f(r)-5 b(e)g(curr)g(enc)g(e)1319 1820 y Fp(u)1375 1835 y Ft(n)p Fi(+2)1539 1820 y Fs(=)28 b Fp(au)1750 1835 y Ft(n)p Fi(+1)1909 1820 y Fs(+)22 b Fp(bu)2104 1835 y Ft(n)2186 1820 y Fv(for)34 b Fp(n)28 b Fk(\025)g Fs(1)739 b(\(5\))456 1999 y Fv(is)34 b(in)h Fk(E)9 b(R)35 b Fv(if)f(and)g(only)h(if)1461 1959 y Ft(u)1502 1968 y Ff(2)p 1461 1976 76 4 v 1461 2033 a Ft(u)1502 2042 y Ff(1)1574 1999 y Fs(=)1687 1960 y Ft(a)1724 1936 y Ff(2)1759 1960 y Fi(+2)p Ft(b)p 1687 1976 193 4 v 1765 2033 a(a)1890 1999 y Fv(.)555 2184 y Fs(As)42 b(an)g(application,)i (Example)p 0 1 0 0 TeXcolorcmyk 44 w(2.2)p Black (#theorem.2.2) [[289 455 304 467] [1 1 1 [3 3]] [0 0 1]] pdfm (.1)d(b) s(ecomes)i(the)f(sharp)s(er)g(result)h(that)456 2300 y(the)36 b(Lucasian)h(sequence)h Fp(a;)17 b(b;)g(a)25 b Fs(+)g Fp(b;)17 b(a)24 b Fs(+)h(2)p Fp(b;)17 b Fs(2)p Fp(a)24 b Fs(+)g(3)p Fp(b;)17 b(:)g(:)g(:)52 b Fs(lies)37 b(in)g Fk(E)9 b(R)35 b Fs(if)h(and)456 2416 y(only)e(if)f Fp(b)c Fs(=)h(3)p Fp(a)p Fs(.)46 b(Moreo)m(v)m(er,)35 b(if)e Fp(f)1704 2431 y Fi(1)1773 2416 y Fs(=)c(1)p Fp(;)17 b(f)2019 2431 y Fi(2)2087 2416 y Fs(=)29 b(1)p Fp(;)17 b(f)2333 2431 y Fi(3)2401 2416 y Fs(=)29 b(2)p Fp(;)17 b(:)g(:)g(:)49 b Fs(is)34 b(the)g(Fib)s(onacci)456 2532 y(sequence,)51 b(then)46 b(an)f(easy)i(consequence)h(of)d(Theorem)p 0 1 0 0 TeXcolorcmyk 47 w(2.5)p Black (#theorem.2.5) [[387 413 402 425] [1 1 1 [3 3]] [0 0 1]] pdfm 45 w(is)h(that)f(for)g(an)m(y)456 2649 y Fp(k)36 b Fk(\025)e Fs(1)i(the)g(sequence)i Fp(f)1365 2664 y Ft(k)1408 2649 y Fp(;)17 b(f)1500 2664 y Ft(k)r Fi(+1)1633 2649 y Fp(;)g(f)1725 2664 y Ft(k)r Fi(+2)1857 2649 y Fp(;)g(:)g(:)g(:)52 b Fs(is)37 b(not)e(in)h Fk(E)9 b(R)p Fs(.)54 b(The)37 b(more)f(general) 456 2765 y(case)f(with)h(square)g(discriminan)m(t,)h Fp(a)e Fs(and)g Fp(a)2118 2729 y Fi(2)2181 2765 y Fs(+)23 b(2)p Fp(b)35 b Fs(ha)m(ving)h(a)e(common)i(factor)456 2881 y(and)c(arbitrary)h Fp(u)1116 2896 y Fi(1)1155 2881 y Fp(;)17 b(u)1255 2896 y Fi(2)1326 2881 y Fs(is)33 b(dealt)g(with)g (in)g([)p 0 1 0 0 TeXcolorcmyk(11)p Black (#cite.puri-thesis) [[316 371 327 383] [1 1 1 [3 3]] [0 0 1]] pdfm (].)456 3009 y(Pro)s(of.)f(First)42 b(assume)i(that)1575 2969 y Ft(u)1616 2978 y Ff(2)p 1575 2986 76 4 v 1575 3044 a Ft(u)1616 3053 y Ff(1)1705 3009 y Fs(=)1835 2970 y Ft(a)1872 2946 y Ff(2)1907 2970 y Fi(+2)p Ft(b)p 1835 2986 193 4 v 1912 3044 a(a)2037 3009 y Fs(.)73 b(Then,)46 b(b)m(y)d(the)g(assumption,)j(the)456 3140 y(sequence)30 b Fp(u)d Fs(is)h(a)f(m)m(ultiple)i(of)e(the)h(sequence)i Fp(a;)17 b(a)2298 3104 y Fi(2)2349 3140 y Fs(+)12 b(2)p Fp(b;)17 b(a)2622 3104 y Fi(3)2673 3140 y Fs(+)12 b(3)p Fp(ab;)17 b(:)g(:)g(:)44 b Fs(whic)m(h)29 b(is)456 3256 y(in)34 b Fk(E)9 b(R)35 b Fs(b)s(ecause)h(the)f(subshift)h(of)e (\014nite)h(t)m(yp)s(e)h(corresp)s(onding)g(to)e(the)h(matrix)456 3288 y Fr(\024)508 3369 y Fp(a)87 b(b)509 3485 y Fs(1)d(0)691 3288 y Fr(\025)776 3428 y Fs(realizes)34 b(it)f(\(and)f(therefore)i(an) m(y)f(m)m(ultiple)h(of)e(it\).)555 3606 y(Con)m(v)m(ersely)-8 b(,)36 b(assume)d(that)f Fp(u)g Fs(is)h(a)e(sequence)k(in)e Fk(E)9 b(R)32 b Fs(satisfying)h(\()p 0 1 0 0 TeXcolorcmyk(5)p Black (#equation.5) [[438 284 444 296] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)43 b(W)-8 b(rite)456 3722 y Fp(x)33 b Fs(for)f(the)h(sequence)1405 3889 y Fp(x)28 b Fs(:)g(2)p Fp(b;)17 b Fs(2)p Fp(ab;)g Fs(2\()p Fp(a)2000 3848 y Fi(2)2040 3889 y Fp(b)22 b Fs(+)g Fp(b)2242 3848 y Fi(2)2282 3889 y Fs(\))p Fp(;)17 b(:)g(:)g(:)456 4056 y Fs(and)32 b Fp(y)k Fs(for)c(the)h(sequence)1436 4223 y Fp(y)e Fs(:)d(2)p Fp(ab;)17 b Fs(2\()p Fp(a)1893 4182 y Fi(2)1932 4223 y Fp(b)23 b Fs(+)f(2)p Fp(b)2184 4182 y Fi(2)2224 4223 y Fs(\))p Fp(;)17 b(:)g(:)g(:)f(;)456 4390 y Fs(b)s(oth)32 b(satisfying)i(the)f(recurrence)h(\()p 0 1 0 0 TeXcolorcmyk(5)p Black (#equation.5) [[288 190 294 202] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)43 b(Notice)33 b(that)1540 4556 y(2)p Fp(bu)1686 4571 y Ft(n)1760 4556 y Fs(=)28 b Fp(Ax)1992 4571 y Ft(n)2061 4556 y Fs(+)22 b Fp(B)5 b(y)2286 4571 y Ft(n)2333 4556 y Fp(;)456 4723 y Fs(for)32 b(in)m(tegers)i Fp(A)e Fs(and)h Fp(B)5 b Fs(.)43 b(By)34 b(\()p 0 1 0 0 TeXcolorcmyk(3)p Black (#equation.3) [[264 150 270 162] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)e(for)g(an)m(y)h(prime)h Fp(p)1265 4890 y(Ax)1393 4905 y Ft(p)1455 4890 y Fs(+)22 b Fp(B)5 b(y)1680 4905 y Ft(p)1747 4890 y Fk(\021)29 b Fp(Ax)1981 4905 y Fi(1)2043 4890 y Fs(+)22 b Fp(B)5 b(y)2268 4905 y Fi(1)2339 4890 y Fs(mo)s(d)33 b Fp(p:)685 b Fs(\(6\))456 5061 y(On)41 b(the)g(other)h(hand,)h(it)e(is)h(w)m(ell-kno)m(wn)h(that)e Fp(x)2351 5076 y Ft(p)2433 5061 y Fk(\021)i Fs(2)p Fp(b)p Fs(\()2691 5022 y Fi(\001)p 2691 5038 59 4 v 2703 5096 a Ft(p)2760 5061 y Fs(\))e(mo)s(d)g Fp(p)g Fs(\(where)456 5202 y(\()504 5162 y Fi(\001)p 504 5179 V 515 5236 a Ft(p)572 5202 y Fs(\))h(is)g(the)h(Legendre)g(sym)m(b)s(ol\),)i(and)d Fp(y)2022 5217 y Ft(p)2105 5202 y Fk(\021)i Fs(2)p Fp(ab)e Fs(mo)s(d)g Fp(p)g Fs(\(b)m(y)h(the)f(previous)p Black Black eop %%Page: 8 8 8 7 bop Black 0 TeXcolorgray Black 456 251 a Fm(8)p Black 456 450 a Fs(paragraph:)43 b Fp(y)35 b Fs(is)e(in)g Fk(E)9 b(R)p Fs(\).)43 b(So)33 b(\()p 0 1 0 0 TeXcolorcmyk(6)p Black (#equation.6) [[274 663 280 675] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(implies)i(that)1336 696 y(2)p Fp(bA)1516 556 y Fr(\022\022)1673 629 y Fs(\001)p 1673 673 82 4 v 1689 764 a Fp(p)1764 556 y Fr(\023)1860 696 y Fk(\000)22 b Fs(1)2008 556 y Fr(\023)2109 696 y Fk(\021)28 b Fs(0)k(mo)s(d)h Fp(p)756 b Fs(\(7\))456 937 y(for)32 b(all)g(primes)i Fp(p)p Fs(.)555 1053 y(W)-8 b(e)24 b(no)m(w)g(claim)g(that)g(the)f (Legendre)i(sym)m(b)s(ol)g(\()2303 1014 y Fi(\001)p 2303 1030 59 4 v 2314 1087 a Ft(p)2371 1053 y Fs(\))f(is)g Fk(\000)p Fs(1)f(for)g(in\014nitely)i(man)m(y)456 1178 y(v)-5 b(alues)34 b(of)f(the)h(prime)g Fp(p)p Fs(.)47 b(This)35 b(completes)g(the)f(pro)s(of)f(of)g(Theorem)p 0 1 0 0 TeXcolorcmyk 34 w(2.5)p Black (#theorem.2.5) [[438 576 453 588] [1 1 1 [3 3]] [0 0 1]] pdfm (,)h (since)456 1294 y(\()p 0 1 0 0 TeXcolorcmyk(7)p Black (#equation.7) [[131 562 137 574] [1 1 1 [3 3]] [0 0 1]] pdfm (\))27 b(forces)h Fp(A)f Fs(=)h(0)f(and)h(hence)g Fp(u)f Fs(is)h(a)f(m)m (ultiple)i(of)2367 1255 y Fi(1)p 2352 1271 66 4 v 2352 1328 a(2)p Ft(b)2428 1294 y Fp(y)t Fs(,)e(namely)i Fp(a;)17 b(a)3014 1258 y Fi(2)3065 1294 y Fs(+)12 b(2)p Fp(b;)17 b(:)g(:)g(:)e Fs(.)555 1411 y(T)-8 b(o)24 b(see)h(the)g(claim,)h(c)m (ho)s(ose)f Fp(c)f Fs(suc)m(h)h(that)f(\()p Fp(c;)17 b Fs(\001\))27 b(=)h(1)23 b(and)i(the)f(Jacobi)g(sym)m(b)s(ol)456 1527 y(\()518 1488 y Ft(c)p 504 1505 59 4 v 504 1562 a Fi(\001)572 1527 y Fs(\))29 b(=)g Fk(\000)p Fs(1.)47 b(Then)34 b(b)m(y)h(Diric)m(hlet,)f(there)h(are)e(in\014nitely)i(man)m (y)g(primes)f Fp(p)g Fs(with)456 1655 y Fp(p)j Fk(\021)h Fp(c)g Fs(mo)s(d)g(\001)g(and)h Fp(p)e Fk(\021)h Fs(1)g(mo)s(d)g(4.)60 b(F)-8 b(or)37 b(suc)m(h)j(primes,)h(\()2741 1611 y Ft(p)p 2729 1632 V 2729 1689 a Fi(\001)2798 1655 y Fs(\))c(=)g(\()3034 1615 y Fi(\001)p 3034 1632 V 3046 1689 a Ft(p)3103 1655 y Fs(\))g(=)g Fk(\000)p Fs(1,)456 1779 y(whic)m(h)d(completes)g(the)f (pro)s(of.)1767 b Fe(2)555 1896 y Fs(The)30 b(case)h(of)d(square)j (discriminan)m(t)g(is)f(m)m(uc)m(h)h(more)e(in)m(v)m(olv)m(ed.)45 b(A)30 b(full)f(treat-)456 2012 y(men)m(t)42 b(is)f(in)h([)p 0 1 0 0 TeXcolorcmyk(11)p Black (#cite.puri-thesis) [[188 476 200 488] [1 1 1 [3 3]] [0 0 1]] pdfm (];)j(here)d(w)m(e)g(simply)h(sho)m(w)f(b)m(y)g(examples)i(that)c(the)i (result)g(as)456 2128 y(stated)33 b(no)f(longer)h(holds)g(in)g (general.)p Black 456 2318 a Fo(Example)38 b(2.6.)p Black Black 190 w Fs(\(1\))p Black 42 w(There)29 b(are)g(in\014nitely)h(man)m (y)g(p)s(ossible)g(v)-5 b(alues)30 b(of)e(the)770 2435 y(ratio)1013 2394 y Ft(u)1054 2403 y Ff(2)p 1013 2412 76 4 v 1013 2469 a Ft(u)1054 2478 y Ff(1)1131 2435 y Fs(for)k(binary)h(recurren)m(t)h(sequences)i(in)d Fk(E)9 b(R)32 b Fs(satisfying)1542 2629 y Fp(u)1598 2644 y Ft(n)p Fi(+2)1763 2629 y Fs(=)27 b Fp(u)1922 2644 y Ft(n)p Fi(+1)2081 2629 y Fs(+)22 b(2)p Fp(u)2284 2644 y Ft(n)2330 2629 y Fp(:)963 b Fs(\(8\))770 2814 y(T)-8 b(o)41 b(see)h(this)f(w)m(e)h (construct)g(t)m(w)m(o)f(di\013eren)m(t)h(realizing)g(examples)g(and) 770 2930 y(then)47 b(tak)m(e)h(linear)f(in)m(tegral)g(com)m(binations)h (of)e(them.)87 b(The)48 b(\014rst)f(is)770 3047 y(the)37 b(subshift)h(of)f(\014nite)g(t)m(yp)s(e)h Fp(T)50 b Fs(corresp)s (onding)38 b(to)e(the)h(matrix)h Fp(A)d Fs(=)770 3078 y Fr(\024)823 3159 y Fs(1)83 b(2)823 3275 y(1)g(0)1003 3078 y Fr(\025)1056 3219 y Fs(.)49 b(This)36 b(system)g(has)f(\(b)m(y)h ([)p 0 1 0 0 TeXcolorcmyk(10)p Black (#cite.MR97a:58050) [[319 331 331 343] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(,)f(Prop)s(osition)g(2.2.12]\))f Fp(f)3144 3234 y Ft(n)3191 3219 y Fs(\()p Fp(T)14 b Fs(\))31 b(=)770 3396 y(trace)q(\()p Fp(A)1093 3360 y Ft(n)1140 3396 y Fs(\),)i(whic)m(h)i(is)f(the)g(sequence)j(of)c(Jacobsthal{Lucas)h(n)m (um)m(b)s(ers)p 0 1 0 0 TeXcolorcmyk 770 3513 a(A014551)p Black [[164 295 208 307] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=014551) pdfm (:1)p Fp(;)17 b Fs(5)p Fp(;)g Fs(7)p Fp(;)g Fs(17)p Fp(;)g(:)g(:)g(:)50 b Fs(\(shifted)38 b(b)m(y)g(one\))g(and)f(has)g(initial)h(ratio)f(5.) 770 3629 y(On)g(the)g(other)g(hand,)i(the)e(algebraic)g(dynamical)h (system)h Fp(S)k Fs(dual)37 b(to)770 3745 y Fp(x)d Fk(7!)f(\000)p Fs(2)p Fp(x)j Fs(on)g(the)g(discrete)i(group)d Fh(Z)p Fs([)2266 3706 y Fi(1)p 2266 3722 36 4 v 2266 3779 a(2)2309 3745 y Fs(])h(has)g(\(see,)i(for)d(example,)j([)p 0 1 0 0 TeXcolorcmyk(3)p Black (#cite.MR99b:11089) [[476 268 482 280] [1 1 1 [3 3]] [0 0 1]] pdfm (,) 770 3862 y(Lemma)i(5.2]\))g Fp(f)1403 3877 y Ft(n)1450 3862 y Fs(\()p Fp(S)6 b Fs(\))39 b(=)h Fk(j)p Fs(\()p Fk(\000)p Fs(2\))1977 3826 y Ft(n)2051 3862 y Fk(\000)27 b Fs(1)p Fk(j)p Fs(,)41 b(whic)m(h)g(b)s(egins)g(3)p Fp(;)17 b Fs(3)p Fp(;)g Fs(9)p Fp(;)g Fs(15)p Fp(;)g(:)g(:)g(:)770 3979 y Fs(and)37 b(has)g(ratio)f(1.)55 b(No)m(w)37 b(w)m(e)h(ma)m(y)g (apply)f(Lemma)p 0 1 0 0 TeXcolorcmyk 37 w(2.3)p Black (#theorem.2.3) [[399 240 414 252] [1 1 1 [3 3]] [0 0 1]] pdfm 36 w(as)g(follo)m(ws.)56 b(If)770 4095 y Fp(s;)17 b(t)33 b Fk(2)g Fh(N)51 b Fs(then)36 b(\()p Fp(tf)1481 4110 y Ft(n)1528 4095 y Fs(\()p Fp(T)14 b Fs(\))22 b(+)g Fp(sf)1889 4110 y Ft(n)1936 4095 y Fs(\()p Fp(S)6 b Fs(\)\))35 b(in)h Fk(E)9 b(R)35 b Fs(is)i(a)e(sequence)j(satisfying)770 4211 y(\()p 0 1 0 0 TeXcolorcmyk(8)p Black (#Item.10) [[169 212 175 224] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)72 b(It)43 b(follo)m(ws)g(that)f(the)g(set)h(of)f(p)s(ossible)i(ratios) 2789 4171 y Ft(u)2830 4180 y Ff(2)p 2789 4188 76 4 v 2789 4245 a Ft(u)2830 4254 y Ff(1)2917 4211 y Fs(con)m(tains)f(the)770 4345 y(in\014nite)34 b(set)f Fk(f)1313 4306 y Fi(5)p Ft(t)p Fi(+3)p Ft(s)p 1313 4322 184 4 v 1330 4379 a(t)p Fi(+3)p Ft(s)1534 4345 y Fk(j)27 b Fp(s;)17 b(t)28 b Fk(2)g Fh(N)9 b Fk(g)p Fs(.)p Black 604 4469 a(\(2\))p Black 41 w(A)35 b(simpler)g(example)h(is)f(giv)m(en)h(b)m(y)f(the)g (Mersenne)h(recurrence.)51 b(Since)770 4586 y(\(2)857 4549 y Ft(n)904 4586 y Fs(\))46 b(and)g(\(1\))g(are)g(b)s(oth)g(in)g Fk(E)9 b(R)p Fs(,)50 b(for)45 b(an)m(y)i Fp(t;)17 b(s)51 b Fk(\025)g Fs(0)46 b(the)h(sequence)770 4702 y(\()p Fp(t)p Fs(2)892 4666 y Ft(n)961 4702 y Fs(+)22 b Fp(s)p Fs(\))33 b(satisfying)g(the)g(recurrence)1531 4886 y Fp(u)1587 4901 y Ft(n)p Fi(+2)1751 4886 y Fs(=)28 b(3)p Fp(u)1960 4901 y Ft(n)p Fi(+1)2118 4886 y Fk(\000)23 b Fs(2)p Fp(u)2323 4901 y Ft(n)3320 4886 y Fs(\(9\))770 5071 y(is)j(in)f Fk(E)9 b(R)p Fs(.)41 b(Th)m(us)27 b(the)e(set)h(of)e (p)s(ossible)j(ratios)2463 5031 y Ft(u)2504 5040 y Ff(2)p 2463 5048 76 4 v 2463 5105 a Ft(u)2504 5114 y Ff(1)2573 5071 y Fs(for)e(exactly)h(realizable)770 5208 y(solutions)34 b(of)e(\()p 0 1 0 0 TeXcolorcmyk(9)p Black (#Item.12) [[231 92 237 104] [1 1 1 [3 3]] [0 0 1]] pdfm (\))g(con)m (tains)i(the)f(in\014nite)g(set)2479 5127 y Fr(\010)2548 5168 y Fi(4)p Ft(t)p Fi(+)p Ft(s)p 2548 5185 149 4 v 2548 5242 a Fi(2)p Ft(t)p Fi(+)p Ft(s)2733 5208 y Fk(j)28 b Fp(s;)17 b(t)27 b Fk(2)h Fh(N)3101 5127 y Fr(\011)3182 5208 y Fp(:)p Black Black eop %%Page: 9 9 9 8 bop Black 0 TeXcolorgray Black 3406 251 a Fm(9)p Black 555 450 a Fs(F)-8 b(or)24 b(higher)h(order)g(recurrences)i(with)e (companion)g(p)s(olynomials)h(irreducible)456 566 y(o)m(v)m(er)k(the)g (rationals,)g(it)f(is)h(clear)g(that)f(some)i(analogue)e(of)g(Theorem)p 0 1 0 0 TeXcolorcmyk 31 w(2.5)p Black (#theorem.2.5) [[437 649 452 661] [1 1 1 [3 3]] [0 0 1]] pdfm 29 w(holds.)456 683 y(The)24 b(rational)f(solutions)h(of)f(a)g Fp(k)s Fs(th)g(order)h(reccurence)i(form)d(a)g(rational)g Fp(k)s Fs(-space;)456 799 y(the)36 b(smallest)i(subspace)g(con)m (tained)f(in)f Fk(E)9 b(R)36 b Fs(has)g(dimension)i(strictly)f(smaller) 456 915 y(than)32 b Fp(k)s Fs(.)44 b(Is)33 b(this)g(dimension)h(alw)m (a)m(ys)h(1?)1392 1113 y(3.)49 b Fl(Realiza)-7 b(tion)38 b(in)g(ra)-7 b(te)555 1287 y Fs(W)f(rite)41 b Fk(b)p Fp(x)p Fk(c)h Fs(for)f(the)g(greatest)g(in)m(teger)h(less)g(than)f(or)g (equal)h(to)e Fp(x)h Fs(and)g Fk(d)p Fp(x)p Fk(e)456 1403 y Fs(for)35 b(the)h(smallest)h(in)m(teger)f(greater)g(than)f(or)h (equal)g(to)f Fp(x)p Fs(.)53 b(In)36 b(this)h(section)f(w)m(e)456 1519 y(assume)45 b(that)g(sequences)i(are)d(nev)m(er)i(zero.)79 b(Di\013eren)m(t)45 b(complications)h(arise)456 1636 y(from)32 b(zeros)h(of)f(sequences)k(and)d(these)h(are)e(discussed)k (in)c(detail)h(in)g([)p 0 1 0 0 TeXcolorcmyk(11)p Black (#cite.puri-thesis) [[436 521 448 533] [1 1 1 [3 3]] [0 0 1]] pdfm (].)p Black 456 1793 a Fo(Theorem)38 b(3.1.)p Black 42 w Fv(L)-5 b(et)35 b Fp(\013)q(;)17 b(\014)40 b Fv(b)-5 b(e)34 b(p)-5 b(ositive)34 b(c)-5 b(onstants.)p Black 604 1931 a Fs(\(1\))p Black 41 w Fv(If)34 b Fp(\036)930 1946 y Ft(n)1005 1931 y Fk(!)27 b(1)35 b Fv(with)1488 1887 y Ft(\036)1530 1895 y Fd(n)p 1488 1908 85 4 v 1509 1965 a Ft(n)1610 1931 y Fk(!)28 b Fs(0)p Fv(,)34 b(then)h Fp(\036)k(=)-61 b Fk(2)28 b(RR)q Fv(.)p Black 604 2048 a Fs(\(2\))p Black 41 w Fv(The)34 b(se)-5 b(quenc)g(e)34 b Fs(\()p Fk(b)p Fp(n)1505 2012 y Ft(\013)1555 2048 y Fk(c)p Fs(\))28 b Fk(2)g(RR)36 b Fv(if)f(and)f(only)h(if)f Fp(\013)29 b(>)e Fs(1)p Fv(.)p Black 604 2164 a Fs(\(3\))p Black 41 w Fv(The)34 b(se)-5 b(quenc)g(e)34 b Fs(\()p Fk(b)p Fp(\014)1508 2128 y Ft(n)1555 2164 y Fk(c)p Fs(\))28 b Fk(2)g(RR)36 b Fv(if)f(and)f(only)h(if)f Fp(\014)f Fk(\025)c Fs(1)p Fv(.)456 2321 y Fs(Pro)s(of.)j(1.)39 b(Assume)24 b(that)e Fp(\036)27 b Fk(2)h(RR)23 b Fs(and)f(let)h Fp(f)32 b Fs(b)s(e)22 b(the)h(corresp)s(onding)g(sequence)456 2439 y(of)35 b(p)s(erio)s(dic)h(p)s(oin)m(ts.)52 b(Then)1558 2395 y Ft(f)1592 2403 y Fd(n)p 1554 2416 V 1554 2473 a Ft(\036)1596 2481 y Fd(n)1681 2439 y Fk(!)32 b Fs(1,)k(so)g Fk(f)2112 2395 y Ft(f)2146 2403 y Fd(n)p 2108 2416 V 2108 2473 a Ft(\036)2150 2481 y Fd(n)2202 2439 y Fk(g)g Fs(is)g(b)s(ounded.)53 b(It)35 b(follo)m(ws)i(that)456 2589 y Fk(f)520 2541 y Ft(f)561 2518 y Fg(\003)554 2558 y Fd(n)p 515 2566 V 515 2623 a Ft(\036)557 2631 y Fd(n)658 2589 y Fs(=)812 2550 y Ft(n)p 791 2566 V 791 2623 a(\036)833 2631 y Fd(n)886 2589 y Fp(f)945 2553 y Ft(o)934 2613 y(n)983 2589 y Fk(g)44 b Fs(is)h(b)s(ounded,)j(and)c(hence)i Fp(f)2170 2553 y Fq(\003)2159 2613 y Ft(n)2257 2589 y Fs(=)i Fp(nf)2498 2553 y Ft(o)2487 2613 y(n)2584 2589 y Fs(=)g(0)c(for)f(all)i(large)f Fp(n)p Fs(.)456 2734 y(This)31 b(implies)h(that)f Fp(f)1263 2749 y Ft(n)1340 2734 y Fs(is)h(b)s(ounded,)f(and)g(so)2180 2690 y Ft(f)2214 2698 y Fd(n)p 2176 2711 V 2176 2769 a Ft(\036)2218 2777 y Fd(n)2298 2734 y Fk(!)c Fs(0,)k(whic)m(h)h(con)m(tradicts)g(the)456 2856 y(assumption.)456 2972 y(2.)81 b(F)-8 b(or)44 b Fp(\013)50 b Fk(2)f Fs(\(0)p Fp(;)17 b Fs(1\))44 b(this)i(follo)m(ws)g (from)f(part)g(1.)81 b(Supp)s(ose)46 b(therefore)g(that)456 3088 y(\()p Fp(n)p Fs(\))30 b Fk(2)i(RR)p Fs(.)49 b(Then)36 b(there)f(is)g(a)f(sequence)j Fp(f)k Fk(2)31 b(E)9 b(R)34 b Fs(with)h Fp(f)2697 3103 y Ft(n)2744 3088 y Fp(=n)c Fk(!)f Fs(1,)35 b(so)g(for)e Fp(p)456 3205 y Fs(a)i(prime,)j Fp(pf)956 3168 y Ft(o)945 3229 y(p)1027 3205 y Fs(=)33 b Fp(f)1195 3168 y Fq(\003)1184 3229 y Ft(p)1267 3205 y Fs(=)g Fp(f)1424 3220 y Ft(p)1488 3205 y Fk(\000)25 b Fp(f)1649 3168 y Fq(\003)1638 3229 y Fi(1)1688 3205 y Fs(,)37 b(and)f(therefore)g Fp(f)2416 3168 y Ft(o)2405 3229 y(p)2487 3205 y Fk(!)d Fs(1)j(as)g Fp(p)d Fk(!)f(1)p Fs(.)53 b(Since)456 3321 y Fp(f)515 3285 y Ft(o)504 3346 y(p)592 3321 y Fs(is)41 b(an)f(in)m(teger,)j(it)d(follo)m(ws)g(that)g Fp(f)1910 3285 y Ft(o)1899 3346 y(p)1988 3321 y Fs(=)g(1)g(for)f(all)h (large)g Fp(p)p Fs(.)66 b(No)m(w)40 b(let)g Fp(q)k Fs(b)s(e)456 3437 y(another)32 b(large)h(prime.)44 b(Then)967 3584 y Fp(f)1015 3599 y Ft(pq)p 967 3628 122 4 v 980 3719 a Fp(pq)1127 3651 y Fs(=)1240 3575 y Fp(f)1299 3538 y Fq(\003)1288 3599 y Ft(pq)1384 3575 y Fs(+)22 b Fp(f)1541 3538 y Fq(\003)1530 3599 y Ft(p)1602 3575 y Fs(+)g Fp(f)1759 3538 y Fq(\003)1748 3599 y Ft(q)1821 3575 y Fs(+)g Fp(f)1978 3538 y Fq(\003)1967 3599 y Fi(1)p 1240 3628 777 4 v 1581 3719 a Fp(pq)2054 3651 y Fs(=)2168 3575 y Fp(f)2227 3538 y Fq(\003)2216 3599 y Ft(pq)p 2168 3628 122 4 v 2181 3719 a Fp(pq)2322 3651 y Fs(+)2430 3584 y(1)p 2430 3628 49 4 v 2430 3719 a Fp(p)2511 3651 y Fs(+)2619 3584 y(1)p 2619 3628 V 2620 3719 a Fp(q)2699 3651 y Fs(+)2807 3584 y Fp(f)2866 3547 y Fq(\003)2855 3608 y Fi(1)p 2807 3628 99 4 v 2808 3719 a Fp(pq)2916 3651 y(;)456 3831 y Fs(so)1473 3908 y(1)p 1473 3953 49 4 v 1473 4044 a Fp(p)1554 3976 y Fs(+)1662 3908 y(1)p 1662 3953 V 1663 4044 a Fp(q)1743 3976 y Fs(+)1851 3908 y Fp(f)1910 3872 y Fq(\003)1899 3933 y Fi(1)p 1851 3953 99 4 v 1852 4044 a Fp(pq)1981 3976 y Fk(\000)2090 3908 y Fp(f)2138 3923 y Ft(pq)p 2090 3953 122 4 v 2103 4044 a Fp(pq)2250 3976 y Fk(2)28 b Fh(Z)p Fp(:)456 4161 y Fs(Fix)k Fp(p)h Fs(large)f(and)h(let)g Fp(q)j Fs(tend)d(to)g(in\014nit)m(y)h(to)e(see)i(that)1818 4294 y(1)p 1818 4338 49 4 v 1818 4429 a Fp(p)1904 4361 y Fk(2)29 b Fh(Z)p Fp(;)456 4572 y Fs(whic)m(h)i(is)g(imp)s(ossible.)45 b(The)31 b(same)g(argumen)m(t)g(sho)m(ws)h(that)e Fp(f)2750 4587 y Ft(n)2797 4572 y Fp(=n)h Fs(cannot)f(ha)m(v)m(e)456 4689 y(an)m(y)j(p)s(ositiv)m(e)h(limit)f(as)g Fp(n)28 b Fk(!)f(1)p Fs(.)555 4805 y(F)-8 b(or)38 b Fp(\013)g(>)f Fs(1,)j(let)f Fp(f)1271 4769 y Ft(o)1260 4830 y(n)1346 4805 y Fs(=)e Fk(d)p Fp(n)1561 4769 y Ft(\013)p Fq(\000)p Fi(1)1718 4730 y Fr(Q)1812 4834 y Ft(p)p Fq(j)p Ft(d)1908 4805 y Fs(\(1)26 b Fk(\000)g Fp(p)2173 4769 y Fq(\000)p Ft(\013)2278 4805 y Fs(\))p Fk(e)p Fs(,)40 b(where)f(the)g(pro)s(duct)g (runs)456 4928 y(o)m(v)m(er)33 b(prime)h(divisors)g(only)-8 b(.)44 b(Then)803 5006 y Fr(X)826 5222 y Ft(d)p Fq(j)p Ft(n)964 5101 y Fp(d)1015 5060 y Ft(\013)1081 5006 y Fr(Y)1099 5222 y Ft(p)p Fq(j)p Ft(d)1208 5101 y Fs(\(1)22 b Fk(\000)g Fp(p)1465 5060 y Fq(\000)p Ft(\013)1570 5101 y Fs(\))27 b(=)h Fp(n)1797 5060 y Ft(\013)1874 5101 y Fk(\024)1979 5006 y Fr(X)2002 5222 y Ft(d)p Fq(j)p Ft(n)2140 5101 y Fp(d)-16 b(f)2234 5060 y Ft(o)2223 5126 y(d)2299 5101 y Fs(=)28 b Fp(f)2451 5116 y Ft(n)2525 5101 y Fk(\024)g Fp(n)2688 5060 y Ft(\013)2760 5101 y Fs(+)2858 5006 y Fr(X)2881 5222 y Ft(d)p Fq(j)p Ft(n)3019 5101 y Fp(d;)p Black Black eop %%Page: 10 10 10 9 bop Black 0 TeXcolorgray Black 456 251 a Fm(10)p Black 456 450 a Fs(so)32 b(0)c Fk(\024)g Fp(f)805 465 y Ft(n)874 450 y Fk(\000)23 b Fp(\036)1032 465 y Ft(n)1106 450 y Fk(\024)28 b Fp(o)p Fs(\()p Fp(n)1354 414 y Ft(\013)1404 450 y Fs(\))p Fp(:)456 566 y Fs(3.)50 b(This)36 b(is)f(clear:)49 b(for)34 b Fp(\014)j(<)31 b Fs(1)k(the)g(sequence)j(is)d(ev)m(en)m (tually)j(0;)d(for)g Fp(\014)i(>)31 b Fs(1)k(the)456 683 y(construction)e(used)h(in)f(part)f(2.)43 b(w)m(orks.)1419 b Fe(2)555 799 y Fs(There)43 b(are)f(sequences)k(gro)m(wing)c(more)g (slo)m(wly)i(than)e Fp(n)2697 763 y Ft(\013)2788 799 y Fs(in)g Fk(RR)q Fs(:)62 b(in)43 b([)p 0 1 0 0 TeXcolorcmyk(11)p Black (#cite.puri-thesis) [[470 621 482 633] [1 1 1 [3 3]] [0 0 1]] pdfm -1 w(,)456 915 y(Chap.)62 b(5])39 b(it)f(is)h(sho)m(wn)h(that)f(\()p Fk(b)p Fp(C)7 b(n)1831 879 y Ft(s)1868 915 y Fs(\(log)18 b Fp(n)p Fs(\))2145 879 y Ft(r)2183 915 y Fk(c)p Fs(\))38 b Fk(2)h(RR)g Fs(for)f(an)m(y)i Fp(r)g Fk(\025)f Fs(1)p Fp(;)17 b(C)45 b(>)456 1031 y Fs(0)p Fp(;)17 b(s)27 b Fk(\025)h Fs(1.)902 1322 y(4.)49 b Fl(Comp)-7 b(aring)39 b(orbits)e(with)h(periodic)f(points)555 1496 y Fs(As)f(is)h(w)m (ell-kno)m(wn,)h(if)e Fp(f)1493 1460 y Fq(\003)1568 1496 y Fs(gro)m(ws)g(fast)g(enough,)h(then)f Fp(f)46 b Fs(gro)m(ws)37 b(v)m(ery)g(m)m(uc)m(h)456 1612 y(lik)m(e)30 b Fp(f)690 1576 y Fq(\003)758 1612 y Fs(\(though)e(not)h(con)m(v)m(ersely)j(in)d (the)g(case)h(of)e(sup)s(er-exp)s(onen)m(tial)j(gro)m(wth:)456 1728 y(cf.)84 b(Theorem)p 0 1 0 0 TeXcolorcmyk 47 w(4.2)p Black (#theorem.4.2) [[200 510 215 522] [1 1 1 [3 3]] [0 0 1]] pdfm 46 w(b)s(elo)m(w\).)h(Throughout)46 b(this)h(section)g Fp(f)2775 1743 y Ft(n)2873 1728 y Fs(=)k Fp(f)3048 1743 y Ft(n)3095 1728 y Fs(\()p Fp(T)14 b Fs(\))45 b(and)456 1845 y Fp(f)515 1808 y Fq(\003)504 1869 y Ft(n)581 1845 y Fs(=)28 b Fp(f)744 1808 y Fq(\003)733 1869 y Ft(n)783 1845 y Fs(\()p Fp(T)14 b Fs(\))32 b(for)g(some)i(map)e Fp(T)14 b Fs(.)p Black 456 2037 a Fo(Remark)36 b(4.1.)p Black 40 w Fs(That)31 b Fp(f)1394 2001 y Fq(\003)1383 2062 y Ft(n)1464 2037 y Fs(is)h(close)g(to)e Fp(f)1958 2052 y Ft(n)2036 2037 y Fs(when)i Fp(f)2337 2052 y Ft(n)2415 2037 y Fs(is)f(gro)m(wing)g(exp)s(onen)m(tially)456 2154 y(has)36 b(b)s(een)g(commen)m(ted)i(on)e(b)m(y)h(Lind)f(in)g([)p 0 1 0 0 TeXcolorcmyk(8)p Black (#cite.MR84g:28017) [[317 459 323 471] [1 1 1 [3 3]] [0 0 1]] pdfm (,) g(Sect.)54 b(4].)f(He)37 b(p)s(oin)m(ts)f(out,)h(using)456 2270 y(\()p 0 1 0 0 TeXcolorcmyk(2)p Black (#equation.2) [[131 445 137 457] [1 1 1 [3 3]] [0 0 1]] pdfm (\),)32 b(that)g(if)g Fp(T)46 b Fs(is)33 b(the)g(automorphism)g(of)f(the)h (2-torus)f(corresp)s(onding)h(to)f(the)456 2448 y(matrix)782 2307 y Fr(\024)834 2388 y Fs(2)83 b(1)834 2504 y(1)g(1)1015 2307 y Fr(\025)1109 2448 y Fs(then)42 b Fp(f)1399 2411 y Fq(\003)1388 2472 y Fi(20)1463 2448 y Fs(\()p Fp(T)14 b Fs(\))41 b(is)i(only)f(0)p Fp(:)p Fs(006\045)f(smaller)i(than)e Fp(f)2957 2463 y Fi(20)3032 2448 y Fs(\()p Fp(T)14 b Fs(\).)70 b(The)456 2620 y(sequence)35 b Fp(f)43 b Fs(of)32 b(p)s(erio)s(dic)h(p)s(oin)m(ts)g(for)f(this)i(map)e(is)p 0 1 0 0 TeXcolorcmyk 33 w(A004146)p Black [[358 403 402 415] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=004146) pdfm (.)p Black 456 2889 a Fo(Theorem)38 b(4.2.)p Black Black 190 w Fs(\(1\))p Black 41 w Fv(If)1588 2850 y Fi(1)p 1585 2866 43 4 v 1585 2924 a Ft(n)1654 2889 y Fs(log)18 b Fp(f)1856 2853 y Fq(\003)1845 2914 y Ft(n)1944 2889 y Fk(!)48 b Fp(C)56 b Fk(2)49 b Fs([0)p Fp(;)17 b Fk(1)p Fs(])45 b Fv(then)2867 2850 y Fi(1)p 2863 2866 V 2863 2924 a Ft(n)2933 2889 y Fs(log)17 b Fp(f)3123 2904 y Ft(n)3219 2889 y Fk(!)48 b Fp(C)770 3005 y Fv(also.)p Black 604 3122 a Fs(\(2\))p Black 784 3082 a Fi(1)p 780 3099 V 780 3156 a Ft(n)850 3122 y Fs(log)17 b Fp(f)1051 3086 y Fq(\003)1040 3146 y Ft(n)1118 3122 y Fk(!)27 b Fp(C)35 b Fk(2)28 b Fs(\(0)p Fp(;)17 b Fk(1)p Fs(\))34 b Fv(if)g(and)h(only)f(if)2351 3082 y Fi(1)p 2347 3099 V 2347 3156 a Ft(n)2416 3122 y Fs(log)18 b Fp(f)2607 3137 y Ft(n)2682 3122 y Fk(!)27 b Fp(C)7 b Fv(.)p Black 604 3249 a Fs(\(3\))p Black 41 w Fv(If)889 3210 y Fi(1)p 885 3226 V 885 3283 a Ft(n)954 3249 y Fs(log)18 b Fp(f)1145 3264 y Ft(n)1224 3249 y Fk(!)31 b(1)37 b Fv(then)1711 3168 y Fr(\010)1783 3210 y Fi(1)p 1779 3226 V 1779 3283 a Ft(n)1849 3249 y Fs(log)17 b Fp(f)2050 3213 y Fq(\003)2039 3274 y Ft(n)2089 3168 y Fr(\011)2185 3249 y Fv(may)37 b(b)-5 b(e)37 b(unb)-5 b(ounde)g(d)36 b(with)h(in\014n-)770 3365 y(itely)e(many)g(limit)f(p)-5 b(oints.)456 3558 y Fs(Pro)s(of.)32 b(1.)42 b(If)976 3519 y Fi(1)p 972 3535 V 972 3593 a Ft(n)1041 3558 y Fs(log)18 b Fp(f)1243 3522 y Fq(\003)1232 3583 y Ft(n)1310 3558 y Fk(!)27 b(1)i Fs(then)1798 3519 y Fi(1)p 1794 3535 V 1794 3593 a Ft(n)1864 3558 y Fs(log)17 b Fp(f)2054 3573 y Ft(n)2129 3558 y Fk(!)27 b(1)i Fs(also,)h(since)h Fp(f)2889 3573 y Ft(n)2963 3558 y Fk(\025)e Fp(f)3128 3522 y Fq(\003)3117 3583 y Ft(n)3196 3558 y Fs(for)f(all)456 3683 y Fp(n)p Fs(.)43 b(If)696 3644 y Fi(1)p 692 3660 V 692 3717 a Ft(n)761 3683 y Fs(log)18 b Fp(f)963 3647 y Fq(\003)952 3707 y Ft(n)1030 3683 y Fk(!)27 b Fp(C)35 b Fk(2)28 b Fs([0)p Fp(;)17 b Fk(1)p Fs(\),)31 b(then)j(\(for)e Fp(n)g Fs(large)h(enough)g (to)f(ha)m(v)m(e)i Fp(f)3150 3647 y Fq(\003)3139 3707 y Ft(n)3217 3683 y Fk(6)p Fs(=)27 b(0\))913 3935 y(1)p 908 3980 59 4 v 908 4071 a Fp(n)993 4003 y Fs(log)17 b Fp(f)1194 3962 y Fq(\003)1183 4028 y Ft(n)1261 4003 y Fk(\024)1381 3935 y Fs(1)p 1376 3980 V 1376 4071 a Fp(n)1461 4003 y Fs(log)h Fp(f)1652 4018 y Ft(n)1783 4003 y Fs(=)1957 3935 y(1)p 1952 3980 V 1952 4071 a Fp(n)2037 4003 y Fs(log)2180 3803 y Fr(0)2180 3982 y(@)2267 3908 y(X)2290 4124 y Ft(d)p Fq(j)p Ft(n)2427 4003 y Fp(f)2486 3962 y Fq(\003)2475 4028 y Ft(d)2526 3803 y Fr(1)2526 3982 y(A)1782 4326 y Fk(\024)1957 4259 y Fs(1)p 1952 4303 V 1952 4395 a Fp(n)2037 4326 y Fs(log)g Fp(n)k Fs(+)2373 4259 y(1)p 2368 4303 V 2368 4395 a Fp(n)2453 4326 y Fs(log)17 b(max)2637 4393 y Ft(d)p Fq(j)p Ft(n)2777 4326 y Fk(f)p Fp(f)2886 4285 y Fq(\003)2875 4351 y Ft(d)2925 4326 y Fk(g)p Fp(:)456 4575 y Fs(F)-8 b(or)40 b(eac)m(h)h(suc)m(h)i Fp(n)p Fs(,)g(c)m(ho)s(ose)j(~)-54 b Fp(n)42 b Fk(2)g(f)p Fp(d)g Fk(j)f Fp(d)p Fk(j)p Fp(n;)17 b(f)2200 4539 y Fq(\003)2189 4601 y Ft(d)2280 4575 y Fk(\025)42 b Fp(f)2458 4539 y Fq(\003)2447 4602 y Ft(d)2483 4583 y Fg(0)2543 4575 y Fk(8)33 b Fp(d)2682 4539 y Fq(0)2705 4575 y Fk(j)p Fp(n)p Fk(g)40 b Fs(so)h(that)g Fp(f)3288 4539 y Fq(\003)3281 4600 y Fi(~)-39 b Ft(n)3369 4575 y Fs(=)456 4694 y(max)637 4709 y Ft(d)p Fq(j)p Ft(n)740 4694 y Fk(f)p Fp(f)849 4658 y Fq(\003)838 4720 y Ft(d)888 4694 y Fk(g)32 b Fs(and)1174 4655 y Fi(~)-39 b Ft(n)p 1170 4671 43 4 v 1170 4728 a(n)1250 4694 y Fk(\024)29 b Fs(1.)43 b(Then)1174 4876 y(1)p 1170 4920 59 4 v 1170 5011 a Fp(n)1254 4943 y Fs(log)18 b Fp(f)1445 4958 y Ft(n)1575 4943 y Fk(\024)1750 4876 y Fs(1)p 1746 4920 V 1746 5011 a Fp(n)1830 4943 y Fs(log)g Fp(n)k Fs(+)2166 4876 y(~)-54 b Fp(n)p 2161 4920 V 2161 5011 a(n)2252 4943 y Fk(\001)2316 4876 y Fs(1)p 2311 4920 V 2316 5011 a(~)g Fp(n)2396 4943 y Fs(log)18 b Fp(f)2598 4902 y Fq(\003)2591 4968 y Fi(~)-39 b Ft(n)1575 5180 y Fk(\024)1750 5113 y Fs(1)p 1746 5158 V 1746 5249 a Fp(n)1830 5180 y Fs(log)18 b Fp(n)k Fs(+)2166 5113 y(1)p 2161 5158 V 2166 5249 a(~)-54 b Fp(n)2246 5180 y Fs(log)18 b Fp(f)2448 5139 y Fq(\003)2441 5205 y Fi(~)-39 b Ft(n)2514 5180 y Fk(!)28 b Fp(C)r(:)p Black Black eop %%Page: 11 11 11 10 bop Black 0 TeXcolorgray Black 3368 251 a Fm(11)p Black 456 450 a Fs(2.)40 b(It)24 b(is)h(enough)g(to)f(sho)m(w)h(that)f (if)1729 411 y Fi(1)p 1726 427 43 4 v 1726 484 a Ft(n)1795 450 y Fs(log)18 b Fp(f)1986 465 y Ft(n)2060 450 y Fk(!)28 b Fp(C)34 b Fk(2)28 b Fs(\(0)p Fp(;)17 b Fk(1)p Fs(\))23 b(then)2906 411 y Fi(1)p 2902 427 V 2902 484 a Ft(n)2972 450 y Fs(log)17 b Fp(f)3173 414 y Fq(\003)3162 475 y Ft(n)3240 450 y Fk(!)27 b Fp(C)456 566 y Fs(also.)43 b(F)-8 b(or)32 b Fp(r)e Fk(\025)e Fs(1,)1052 748 y Fp(f)1100 763 y Ft(r)1165 748 y Fk(\025)g Fp(f)1329 707 y Fq(\003)1318 773 y Ft(r)1396 748 y Fs(=)g Fk(\000)1637 654 y Fr(X)1594 870 y Ft(d)p Fq(j)p Ft(r)n(;d)p Fq(6)p Fi(=)p Ft(r)1841 748 y Fp(f)1900 707 y Fq(\003)1889 773 y Ft(d)1961 748 y Fs(+)22 b Fp(f)2107 763 y Ft(r)2173 748 y Fk(\025)28 b Fp(f)2326 763 y Ft(r)2386 748 y Fk(\000)2529 654 y Fr(X)2486 870 y Ft(d)p Fq(j)p Ft(r)n(;d)p Fq(6)p Fi(=)p Ft(r)2733 748 y Fp(f)2781 763 y Ft(d)2821 748 y Fp(:)456 1036 y Fs(Let)34 b Fp(R)g Fs(b)s(e)g(an)g(upp)s(er)g(b)s(ound)g(for)f Fk(f)1807 997 y Fi(1)p 1803 1013 V 1803 1070 a Ft(n)1873 1036 y Fs(log)17 b Fp(f)2063 1051 y Ft(n)2140 1036 y Fk(j)29 b Fp(f)2245 1051 y Ft(n)2322 1036 y Fk(6)p Fs(=)h(0)p Fk(g)j Fs(and)h(pic)m(k)h Fp(\017)30 b Fk(2)g Fs(\(0)p Fp(;)17 b Fs(3)p Fp(C)7 b Fs(\).)456 1152 y(Cho)s(ose)33 b Fp(N)43 b Fs(so)33 b(that)1216 1318 y Fp(r)d(>)e(N)66 b Fs(=)-17 b Fk(\))55 b Fp(e)1797 1277 y Ft(r)r Fi(\()p Ft(C)5 b Fq(\000)p Ft(\017)p Fi(\))2056 1318 y Fk(\024)28 b Fp(f)2209 1333 y Ft(r)2275 1318 y Fk(\024)g Fp(e)2425 1277 y Ft(r)r Fi(\()p Ft(C)5 b Fi(+)p Ft(\017)p Fi(\))2657 1318 y Fp(:)456 1484 y Fs(Then)33 b(for)f Fp(r)f(>)c Fs(2)p Fp(N)43 b Fs(\(so)33 b(that)f Fp(r)1623 1448 y Fq(\003)1662 1484 y Fp(;)17 b(N)38 b Fk(\024)28 b(b)1982 1445 y Ft(r)p 1981 1461 36 4 v 1981 1519 a Fi(2)2027 1484 y Fk(c)p Fs(\),)703 1769 y Fp(f)751 1784 y Ft(r)816 1769 y Fk(\025)h Fp(f)981 1727 y Fq(\003)970 1793 y Ft(r)1103 1769 y Fk(\025)83 b Fp(f)1311 1784 y Ft(r)1371 1769 y Fk(\000)1511 1644 y Ft(N)1471 1674 y Fr(X)1477 1883 y Ft(n)p Fi(=1)1632 1769 y Fp(f)1680 1784 y Ft(n)1749 1769 y Fk(\000)1890 1636 y Fq(b)p Ft(r)r(=)p Fi(2)p Fq(c)1902 1674 y Fr(X)1848 1885 y Ft(n)p Fi(=)p Ft(N)7 b Fi(+1)2116 1769 y Fp(f)2164 1784 y Ft(n)1103 2031 y Fk(\025)83 b Fp(f)1311 2046 y Ft(r)1371 2031 y Fk(\000)1471 1950 y Fr(\000)1517 2031 y Fp(N)10 b(e)1650 1989 y Ft(N)d(R)1793 2031 y Fs(+)22 b(\()p Fp(r)s(=)p Fs(2)g Fk(\000)g Fp(N)10 b Fs(\))p Fp(e)2366 1989 y Ft(r)r Fi(\()p Ft(C)5 b Fi(+)p Ft(\017)p Fi(\))p Ft(=)p Fi(2)2669 1950 y Fr(\001)1103 2194 y Fk(\025)83 b Fp(f)1311 2209 y Ft(r)1366 2113 y Fr(\000)1412 2194 y Fs(1)21 b Fk(\000)i Fp(N)10 b(e)1715 2153 y Ft(N)d(R)p Fq(\000)p Ft(r)r Fi(\()p Ft(C)e Fq(\000)p Ft(\017)p Fi(\))2141 2194 y Fk(\000)22 b Fs(\()p Fp(r)s(=)p Fs(2)g Fk(\000)g Fp(N)10 b Fs(\))p Fp(e)2715 2153 y Fq(\000)p Ft(r)r Fi(\()p Ft(C)5 b Fq(\000)p Fi(3)p Ft(\017)p Fi(\))p Ft(=)p Fi(2)3108 2113 y Fr(\001)3170 2194 y Fp(;)456 2360 y Fs(and)39 b(the)h(brac)m(k)m(eted)h(expression)g(con)m(v)m (erges)h(to)d(1)g(as)g Fp(r)j Fk(!)c(1)p Fs(.)63 b(T)-8 b(aking)40 b(logs)456 2476 y(and)32 b(dividing)i(b)m(y)f Fp(r)j Fs(giv)m(es)e(the)f(result.)456 2592 y(3.)60 b(W)-8 b(rite)39 b Fp(p)918 2607 y Fi(1)957 2592 y Fp(;)17 b(p)1050 2607 y Fi(2)1090 2592 y Fp(;)g(:)g(:)g(:)54 b Fs(for)37 b(the)i(sequence)i(of)d(primes.)63 b(Let)38 b Fp(n)2767 2607 y Ft(r)2843 2592 y Fs(=)f Fp(p)3005 2607 y Ft(r)3043 2592 y Fp(p)3092 2607 y Ft(r)r Fi(+1)3220 2592 y Fs(,)j(and)456 2708 y(de\014ne)46 b(a)e(sequence)k(\()p Fp(f)1357 2672 y Fq(\003)1346 2734 y Ft(k)1396 2708 y Fs(\))d(as)g(follo)m(ws.)81 b(F)-8 b(or)44 b Fp(k)k Fs(not)d(of)f(the)h(form)g Fp(n)3082 2723 y Ft(r)3120 2708 y Fs(,)j(de\014ne)456 2834 y Fp(f)515 2798 y Fq(\003)504 2860 y Ft(k)585 2834 y Fs(=)30 b Fp(k)d Fk(\001)c Fs(2)869 2798 y Ft(k)908 2775 y Ff(3)946 2834 y Fp(:)34 b Fs(F)-8 b(or)34 b Fp(k)j Fs(of)d(the)h(form)f Fp(n)1845 2849 y Ft(r)1917 2834 y Fs(de\014ne)i Fp(f)2260 2798 y Fq(\003)2249 2860 y Ft(k)2333 2834 y Fs(according)f(to)f(the)h (follo)m(wing)456 2950 y(sc)m(heme:)750 3116 y Fp(f)809 3075 y Fq(\003)798 3141 y Ft(n)841 3150 y Ff(1)963 3116 y Fs(=)83 b Fp(n)1180 3131 y Fi(1)1220 3116 y Fs(2)1269 3075 y Ft(n)1312 3084 y Ff(1)750 3272 y Fp(f)809 3230 y Fq(\003)798 3296 y Ft(n)841 3305 y Ff(2)963 3272 y Fs(=)g Fp(n)1180 3287 y Fi(2)1220 3272 y Fs(2)1269 3230 y Ft(n)1312 3239 y Ff(2)1350 3272 y Fp(;)17 b(f)1453 3230 y Fq(\003)1442 3296 y Ft(n)1485 3305 y Ff(3)1551 3272 y Fs(=)27 b Fp(n)1712 3287 y Fi(3)1752 3272 y Fs(2)1801 3230 y Fi(2)p Ft(n)1879 3239 y Ff(3)750 3427 y Fp(f)809 3386 y Fq(\003)798 3451 y Ft(n)841 3460 y Ff(4)963 3427 y Fs(=)83 b Fp(n)1180 3442 y Fi(4)1220 3427 y Fs(2)1269 3386 y Ft(n)1312 3395 y Ff(4)1350 3427 y Fp(;)17 b(f)1453 3386 y Fq(\003)1442 3451 y Ft(n)1485 3460 y Ff(5)1551 3427 y Fs(=)27 b Fp(n)1712 3442 y Fi(5)1752 3427 y Fs(2)1801 3386 y Fi(2)p Ft(n)1879 3395 y Ff(5)1918 3427 y Fp(;)17 b(f)2021 3386 y Fq(\003)2010 3451 y Ft(n)2053 3460 y Ff(6)2119 3427 y Fs(=)27 b Fp(n)2280 3442 y Fi(6)2320 3427 y Fs(2)2369 3386 y Fi(3)p Ft(n)2447 3395 y Ff(6)750 3582 y Fp(f)809 3541 y Fq(\003)798 3606 y Ft(n)841 3615 y Ff(7)963 3582 y Fs(=)83 b Fp(n)1180 3597 y Fi(7)1220 3582 y Fs(2)1269 3541 y Ft(n)1312 3550 y Ff(7)1350 3582 y Fp(;)17 b(f)1453 3541 y Fq(\003)1442 3606 y Ft(n)1485 3615 y Ff(8)1551 3582 y Fs(=)27 b Fp(n)1712 3597 y Fi(8)1752 3582 y Fs(2)1801 3541 y Fi(2)p Ft(n)1879 3550 y Ff(8)1918 3582 y Fp(;)17 b(f)2021 3541 y Fq(\003)2010 3606 y Ft(n)2053 3615 y Ff(9)2119 3582 y Fs(=)27 b Fp(n)2280 3597 y Fi(9)2320 3582 y Fs(2)2369 3541 y Fi(3)p Ft(n)2447 3550 y Ff(9)2486 3582 y Fp(;)17 b(f)2589 3541 y Fq(\003)2578 3606 y Ft(n)2621 3615 y Ff(10)2717 3582 y Fs(=)28 b Fp(n)2879 3597 y Fi(10)2954 3582 y Fs(2)3003 3541 y Fi(4)p Ft(n)3081 3550 y Ff(10)456 3893 y Fs(and)i(so)h(on.)43 b(Then)1200 3854 y Fi(1)p 1197 3870 43 4 v 1197 3927 a Ft(n)1266 3893 y Fs(log)18 b Fp(f)1457 3908 y Ft(n)1531 3893 y Fk(!)28 b(1)i Fs(o\013)g(the)h Fp(n)2149 3908 y Ft(r)2187 3893 y Fs('s)g(clearly)-8 b(.)44 b(Along)31 b(the)g(sequence)456 4010 y(\()p Fp(n)552 4025 y Ft(r)590 4010 y Fs(\),)1185 4136 y Fp(f)1233 4151 y Ft(n)1276 4159 y Fd(r)1342 4136 y Fs(=)d Fp(f)1505 4094 y Fq(\003)1494 4160 y Ft(n)1537 4168 y Fd(r)1598 4136 y Fs(+)22 b Fp(f)1755 4094 y Fq(\003)1744 4160 y Ft(p)1780 4168 y Fd(r)1840 4136 y Fs(+)g Fp(f)1997 4094 y Fq(\003)1986 4160 y Ft(p)2022 4169 y Fd(r)r Ff(+1)2160 4136 y Fs(+)g Fp(f)2317 4094 y Fq(\003)2306 4160 y Fi(1)2384 4136 y Fk(\025)28 b Fp(f)2548 4094 y Fq(\003)2537 4160 y Ft(p)2573 4169 y Fd(r)r Ff(+1)2688 4136 y Fp(;)456 4281 y Fs(so)1068 4363 y(1)p 1045 4407 97 4 v 1045 4498 a Fp(n)1103 4513 y Ft(r)1167 4430 y Fs(log)18 b Fp(f)1358 4445 y Ft(n)1401 4453 y Fd(r)1467 4430 y Fk(\025)1690 4363 y Fs(1)p 1582 4407 265 4 v 1582 4498 a Fp(p)1631 4513 y Ft(r)1669 4498 y Fp(p)1718 4513 y Ft(r)r Fi(+1)1873 4430 y Fs(log)2016 4320 y Fr(\020)2075 4430 y Fp(p)2124 4445 y Ft(r)r Fi(+1)2274 4430 y Fk(\001)k Fs(2)2373 4389 y Ft(p)2409 4366 y Ff(3)2409 4410 y Fd(r)r Ff(+1)2524 4320 y Fr(\021)2611 4430 y Fk(!)28 b(1)p Fp(:)555 4627 y Fs(On)g(the)h(other)f(hand,)h(along)e(a)h(subsequence)k(of)27 b Fp(n)2437 4642 y Ft(r)2475 4627 y Fs('s)i(c)m(hosen)g(to)f(ha)m(v)m (e)h Fp(f)3270 4591 y Fq(\003)3259 4652 y Ft(n)3302 4660 y Fd(r)3369 4627 y Fs(=)456 4751 y Fp(n)514 4766 y Ft(r)552 4751 y Fs(2)601 4714 y Ft(`n)673 4722 y Fd(r)748 4751 y Fs(for)36 b(a)g(\014xed)i Fp(`)d Fk(2)g Fh(N)52 b Fs(\(whic)m(h)38 b(will)g(exist)g(b)m(y)g(construction\),)h(w)m(e)e(realize)456 4867 y Fp(`)17 b Fs(log)g(2)32 b(as)h(a)f(limit)h(p)s(oin)m(t)g(of)f (the)h(sequence)2123 4828 y Fi(1)p 2119 4844 43 4 v 2119 4901 a Ft(n)2189 4867 y Fs(log)17 b Fp(f)2390 4831 y Fq(\003)2379 4892 y Ft(n)2429 4867 y Fs(.)914 b Fe(2)555 4983 y Fs(Finally)-8 b(,)25 b(w)m(e)e(turn)g(to)f(comparing)h(these)g (gro)m(wth)g(rates)g(in)f(a)h(sub-exp)s(onen)m(tial)456 5099 y(setting.)59 b(F)-8 b(or)37 b(p)s(olynomial)h(gro)m(wth,)h(the)f (next)h(result)f(sho)m(ws)i(that)d Fp(f)48 b Fs(and)38 b Fp(f)3405 5063 y Fq(\003)456 5216 y Fs(are)32 b(forced)h(to)f(b)s (eha)m(v)m(e)j(v)m(ery)f(di\013eren)m(tly)-8 b(.)p Black Black eop %%Page: 12 12 12 11 bop Black 0 TeXcolorgray Black 456 251 a Fm(12)p Black Black 456 450 a Fo(Theorem)38 b(4.3.)p Black 42 w Fv(L)-5 b(et)35 b Fp(C)42 b Fv(and)34 b Fp(\013)h Fv(b)-5 b(e)35 b(p)-5 b(ositive)34 b(c)-5 b(onstants.)p Black 604 600 a Fs(\(1\))p Black 41 w Fv(F)e(or)50 b Fp(\013)58 b(>)e Fs(1)p Fv(,)f(the)50 b(set)h Fk(f)1764 552 y Ft(f)1805 529 y Fg(\003)1798 569 y Fd(n)p 1759 577 88 4 v 1759 634 a Ft(n)1802 615 y Fd(\013)1856 600 y Fk(g)f Fv(is)h(b)-5 b(ounde)g(d)50 b(if)g(and)g(only)h(if)f Fk(f)3182 556 y Ft(f)3216 564 y Fd(n)p 3177 577 V 3177 634 a Ft(n)3220 615 y Fd(\013)3274 600 y Fk(g)h Fv(is)770 716 y(b)-5 b(ounde)g(d.)p Black 604 832 a Fs(\(2\))p Black 41 w Fv(F)e(or)34 b Fp(\013)28 b(>)g Fs(1)p Fv(,)1275 788 y Ft(f)1309 796 y Fd(n)p 1269 809 V 1269 867 a Ft(n)1312 848 y Fd(\013)1394 832 y Fk(!)f Fs(0)35 b Fv(if)g(and)f(only)h(if)2210 785 y Ft(f)2251 761 y Fg(\003)2244 801 y Fd(n)p 2205 809 V 2205 867 a Ft(n)2248 848 y Fd(\013)2330 832 y Fk(!)27 b Fs(0)p Fv(.)p Black 604 969 a Fs(\(3\))p Black 41 w Fv(If)886 925 y Ft(f)920 933 y Fd(n)p 881 946 V 881 1003 a Ft(n)924 984 y Fd(\013)1005 969 y Fk(!)h Fp(C)40 b Fv(for)32 b(some)h Fp(\013)28 b(>)f Fs(1)p Fv(,)33 b(then)g Fk(f)2229 921 y Ft(f)2270 898 y Fg(\003)2263 938 y Fd(n)p 2224 946 V 2224 1003 a Ft(n)2267 984 y Fd(\013)2321 969 y Fk(g)g Fv(has)f(in\014nitely)h(many)g(limit)770 1085 y(p)-5 b(oints.)p Black 604 1207 a Fs(\(4\))p Black 41 w Fv(If)885 1160 y Ft(f)926 1136 y Fg(\003)919 1176 y Fd(n)p 880 1184 V 880 1241 a Ft(n)923 1222 y Fd(\013)1005 1207 y Fk(!)27 b Fp(C)40 b Fv(for)33 b(some)f Fp(\013)c Fk(\025)g Fs(1)p Fv(,)33 b(then)g Fk(f)2230 1163 y Ft(f)2264 1171 y Fd(n)p 2225 1184 V 2225 1241 a Ft(n)2268 1222 y Fd(\013)2322 1207 y Fk(g)f Fv(has)h(in\014nitely)f(many)h(limit)770 1323 y(p)-5 b(oints.)456 1512 y Fs(Pro)s(of.)32 b(1.)43 b(Let)33 b Fp(R)g Fs(b)s(e)g(an)f(upp)s(er)i(b)s(ound)e(for)g Fk(f)2213 1464 y Ft(f)2254 1441 y Fg(\003)2247 1481 y Fd(n)p 2208 1489 V 2208 1546 a Ft(n)2251 1527 y Fd(\013)2305 1512 y Fk(g)p Fs(.)43 b(Then)886 1706 y Fp(f)934 1721 y Ft(n)p 879 1750 108 4 v 879 1842 a Fp(n)937 1813 y Ft(\013)1025 1773 y Fk(\024)1169 1706 y Fs(1)p 1140 1750 V 1140 1842 a Fp(n)1198 1813 y Ft(\013)1274 1679 y Fr(X)1297 1894 y Ft(d)p Fq(j)p Ft(n)1435 1773 y Fp(R)q(d)1561 1732 y Ft(\013)1638 1773 y Fs(=)27 b Fp(R)1833 1679 y Fr(X)1855 1894 y Ft(d)p Fq(j)p Ft(n)1993 1633 y Fr(\022)2080 1706 y Fp(d)p 2077 1750 59 4 v 2077 1842 a(n)2145 1633 y Fr(\023)2218 1655 y Ft(\013)2295 1773 y Fk(\024)h Fp(R)2529 1649 y Fq(1)2492 1679 y Fr(X)2501 1891 y Ft(d)p Fi(=1)2688 1706 y Fs(1)p 2663 1750 101 4 v 2663 1842 a Fp(d)2714 1813 y Ft(\013)2800 1773 y Fp(<)g Fk(1)p Fp(:)456 2054 y Fs(The)33 b(con)m(v)m(erse)i(is)e(ob)m(vious.)456 2170 y(2.)69 b(One)42 b(direction)g(is)g(clear.)70 b(Assume)43 b(that)2240 2123 y Ft(f)2281 2100 y Fg(\003)2274 2140 y Fd(n)p 2235 2148 88 4 v 2235 2205 a Ft(n)2278 2186 y Fd(\013)2375 2170 y Fk(!)f Fs(0.)69 b(Fix)42 b Fp(\017)h(>)f Fs(0;)j(c)m(ho)s(ose) 456 2287 y Fp(M)550 2302 y Fi(1)617 2287 y Fk(2)28 b Fh(N)48 b Fs(so)32 b(that)1399 2462 y Fp(n)c(>)g(M)1683 2477 y Fi(1)1778 2462 y Fs(=)-17 b Fk(\))2007 2394 y Fp(f)2066 2358 y Fq(\003)2055 2419 y Ft(n)p 2002 2439 108 4 v 2002 2530 a Fp(n)2060 2501 y Ft(\013)2148 2462 y Fp(<)2356 2394 y(\017)p 2261 2439 230 4 v 2261 2530 a Fs(1)22 b(+)g Fp(\014)456 2675 y Fs(where)33 b Fp(\014)h Fs(=)929 2600 y Fr(P)1034 2627 y Fq(1)1034 2704 y Ft(k)r Fi(=1)1218 2636 y(1)p 1194 2652 83 4 v 1194 2709 a Ft(k)1233 2690 y Fd(\013)1286 2675 y Fs(.)44 b(Cho)s(ose)33 b Fp(M)1790 2690 y Fi(2)1863 2675 y Fs(so)f(that)1306 2954 y Fp(n)c(>)f(M)1589 2969 y Fi(2)1684 2954 y Fs(=)-17 b Fk(\))1919 2829 y Ft(M)1987 2838 y Ff(1)1898 2859 y Fr(X)1908 3068 y Ft(r)r Fi(=1)2074 2886 y Fp(f)2133 2850 y Fq(\003)2122 2911 y Ft(r)p 2069 2931 108 4 v 2069 3022 a Fp(n)2127 2993 y Ft(\013)2214 2954 y Fp(<)2423 2886 y(\017)p 2328 2931 230 4 v 2328 3022 a Fs(1)22 b(+)g Fp(\014)2567 2954 y(:)456 3217 y Fs(Then)33 b(for)f Fp(n)c Fk(\025)g Fs(max)q Fk(f)p Fp(M)1376 3232 y Fi(1)1416 3217 y Fp(;)17 b(M)1554 3232 y Fi(2)1593 3217 y Fk(g)p Fs(,)898 3486 y(0)27 b Fk(\024)1096 3419 y Fp(f)1144 3434 y Ft(n)p 1089 3463 108 4 v 1089 3555 a Fp(n)1147 3526 y Ft(\013)1235 3486 y Fs(=)1338 3392 y Fr(X)1361 3607 y Ft(d)p Fq(j)p Ft(n)1513 3419 y Fp(f)1572 3383 y Fq(\003)1561 3445 y Ft(d)p 1509 3463 V 1509 3555 a Fp(n)1567 3526 y Ft(\013)1709 3486 y Fk(\024)1891 3361 y Ft(M)1959 3370 y Ff(1)1870 3392 y Fr(X)1880 3601 y Ft(r)r Fi(=1)2045 3419 y Fp(f)2104 3383 y Fq(\003)2093 3443 y Ft(r)p 2040 3463 V 2040 3555 a Fp(n)2098 3526 y Ft(\013)2180 3486 y Fs(+)2362 3392 y Fr(X)2278 3607 y Ft(d)p Fq(j)p Ft(n;d>M)2556 3616 y Ff(1)2621 3419 y Fp(f)2680 3383 y Fq(\003)2669 3443 y Ft(r)p 2617 3463 V 2617 3555 a Fp(n)2675 3526 y Ft(\013)1710 3852 y Fs(=)1891 3727 y Ft(M)1959 3736 y Ff(1)1870 3757 y Fr(X)1880 3966 y Ft(r)r Fi(=1)2045 3784 y Fp(f)2104 3748 y Fq(\003)2093 3809 y Ft(r)p 2040 3829 V 2040 3920 a Fp(n)2098 3891 y Ft(\013)2180 3852 y Fs(+)2362 3757 y Fr(X)2278 3973 y Ft(d)p Fq(j)p Ft(n;d>M)2556 3982 y Ff(1)2607 3852 y Fs(+)2697 3784 y Fp(d)2748 3748 y Ft(\013)p 2693 3829 V 2693 3920 a Fp(n)2751 3891 y Ft(\013)2832 3852 y Fk(\001)2893 3784 y Fp(f)2952 3748 y Fq(\003)2941 3810 y Ft(d)p 2892 3829 101 4 v 2892 3920 a Fp(d)2943 3891 y Ft(\013)1709 4173 y Fk(\024)1975 4106 y Fp(\017)p 1880 4150 230 4 v 1880 4242 a Fs(1)22 b(+)g Fp(\014)2141 4173 y Fs(+)2345 4106 y Fp(\017)p 2249 4150 V 2249 4242 a Fs(1)g(+)g Fp(\014)2590 4079 y Fr(X)2506 4295 y Ft(d)p Fq(j)p Ft(n;d>M)2784 4304 y Ff(1)2848 4106 y Fp(d)2899 4070 y Ft(\013)p 2844 4150 108 4 v 2844 4242 a Fp(n)2902 4213 y Ft(\013)1709 4469 y Fk(\024)1975 4401 y Fp(\017)p 1880 4446 230 4 v 1880 4537 a Fs(1)g(+)g Fp(\014)2141 4469 y Fs(+)g Fp(\014)2405 4401 y(\017)p 2309 4446 V 2309 4537 a Fs(1)h(+)f Fp(\014)2577 4469 y Fk(\024)28 b Fp(\017:)456 4724 y Fs(3.)45 b(Assume)35 b(that)1166 4681 y Ft(f)1200 4689 y Fd(n)p 1161 4701 88 4 v 1161 4759 a Ft(n)1204 4740 y Fd(\013)1286 4724 y Fk(!)29 b Fp(C)36 b(>)28 b Fs(0.)45 b(Then)2013 4667 y Ft(f)2054 4644 y Fg(\003)2047 4684 y Fd(p)p 2012 4701 80 4 v 2012 4759 a Ft(p)2048 4740 y Fd(\013)2130 4724 y Fk(!)28 b Fp(C)40 b Fs(along)33 b(primes.)47 b(F)-8 b(or)32 b(a)h(\014xed)456 4846 y(prime)g Fp(p)p Fs(,)1100 4958 y Fp(f)1159 4922 y Fq(\003)1148 4983 y Ft(p)1184 4964 y Fd(r)p 1095 5011 133 4 v 1095 5103 a Fp(p)1144 5074 y Ft(r)r(\013)1265 5034 y Fs(=)1384 4967 y Fp(f)1432 4982 y Ft(p)1468 4963 y Fd(r)p 1379 5011 V 1379 5103 a Fp(p)1428 5074 y Ft(r)r(\013)1543 5034 y Fk(\000)1691 4966 y Fp(f)1739 4983 y Ft(p)1775 4964 y Fd(r)r Fg(\000)p Ff(1)p 1653 5011 278 4 v 1653 5103 a Fp(p)1702 5074 y Fi(\()p Ft(r)r Fq(\000)p Fi(1\))p Ft(\013)1962 5034 y Fk(\001)2047 4967 y Fs(1)p 2022 5011 99 4 v 2022 5103 a Fp(p)2071 5074 y Ft(\013)2158 5034 y Fk(!)2285 4894 y Fr(\022)2359 5034 y Fs(1)22 b Fk(\000)2564 4967 y Fs(1)p 2539 5011 V 2539 5103 a Fp(p)2588 5074 y Ft(\013)2648 4894 y Fr(\023)2738 5034 y Fp(C)456 5216 y Fs(as)32 b Fp(r)f Fk(!)c(1)p Fs(.)p Black Black eop %%Page: 13 13 13 12 bop Black 0 TeXcolorgray Black 3368 251 a Fm(13)p Black 456 462 a Fs(4.)54 b(Assume)39 b(that)1182 415 y Ft(f)1223 391 y Fg(\003)1216 431 y Fd(n)p 1177 439 88 4 v 1177 496 a Ft(n)1220 478 y Fd(\013)1308 462 y Fk(!)34 b Fp(C)41 b(>)34 b Fs(0.)55 b(Then)2068 413 y Ft(f)2102 421 y Fd(p)p 2063 439 80 4 v 2063 496 a Ft(p)2099 478 y Fd(\013)2187 462 y Fk(!)34 b Fp(C)43 b Fs(along)36 b(primes.)57 b(F)-8 b(or)35 b(\014xed)456 587 y(prime)e Fp(p)f Fs(and)h Fp(q)j Fs(prime,)1084 733 y Fp(f)1132 748 y Ft(pq)p 1034 778 222 4 v 1034 869 a Fs(\()p Fp(pq)t Fs(\))1206 840 y Ft(\013)1293 801 y Fs(=)1407 724 y Fp(f)1466 688 y Fq(\003)1455 749 y Ft(pq)1550 724 y Fs(+)22 b Fp(f)1707 688 y Fq(\003)1696 749 y Ft(q)1769 724 y Fs(+)g Fp(f)1926 688 y Fq(\003)1915 749 y Ft(p)1987 724 y Fs(+)g Fp(f)2144 688 y Fq(\003)2133 749 y Fi(1)p 1407 778 777 4 v 1684 869 a Fs(\()p Fp(pq)t Fs(\))1856 840 y Ft(\013)2221 801 y Fk(!)2348 660 y Fr(\022)2421 801 y Fs(1)g(+)2625 733 y(1)p 2600 778 99 4 v 2600 869 a Fp(p)2649 840 y Ft(\013)2709 660 y Fr(\023)2799 801 y Fp(C)456 989 y Fs(as)32 b Fp(q)g Fk(!)27 b(1)p Fs(.)2466 b Fe(2)p Black 456 1166 a Fo(Remark)49 b(4.4.)p Black 47 w Fs(F)-8 b(or)42 b(the)h(case)1703 1118 y Ft(f)1744 1095 y Fg(\003)1737 1135 y Fd(n)p 1703 1143 78 4 v 1720 1200 a Ft(n)1835 1166 y Fk(!)h Fp(C)52 b(>)45 b Fs(0)d(in)h(Theorem)p 0 1 0 0 TeXcolorcmyk 44 w(4.3)p Black (#theorem.4.3) [[415 577 430 589] [1 1 1 [3 3]] [0 0 1]] pdfm (,)3066 1122 y Ft(f)3100 1130 y Fd(n)p 3066 1143 77 4 v 3083 1200 a Ft(n)3195 1166 y Fs(is)g(un-)456 1282 y(b)s(ounded:)h(similar)33 b(argumen)m(ts)h(sho)m(w)g(that)1455 1446 y Fp(f)1503 1461 y Ft(p)1539 1470 y Ff(1)1573 1461 y Ft(p)1609 1470 y Ff(2)1643 1461 y Ft(:::)o(p)1738 1469 y Fd(m)p 1407 1491 441 4 v 1407 1582 a Fp(p)1456 1597 y Fi(1)1496 1582 y Fp(p)1545 1597 y Fi(2)1601 1582 y Fp(:)17 b(:)g(:)f(p)1781 1597 y Ft(m)1885 1514 y Fk(\025)2031 1389 y Ft(m)1990 1419 y Fr(X)2005 1629 y Ft(i)p Fi(=1)2175 1446 y Fs(1)p 2161 1491 78 4 v 2161 1582 a Fp(p)2210 1597 y Ft(i)2276 1514 y Fk(!)27 b(1)456 1731 y Fs(as)32 b Fp(m)c Fk(!)g(1)p Fs(.)1648 1928 y(5.)49 b Fl(Examples)555 2103 y Fs(F)-8 b(ew)34 b(of)e(the)i(standard)g(sequences)i(turn)d(out)g(to)g(b)s(e)h (in)f Fk(E)9 b Fp(R)q Fs(.)45 b(Here)34 b(w)m(e)g(list)g(a)456 2219 y(few)f(that)f(are,)h(and)f(one)h(that)g(nearly)g(is.)44 b(In)33 b(some)g(cases)h(the)f(pro)s(of)f(pro)s(ceeds)456 2335 y(b)m(y)45 b(exhibiting)g(a)f(realizing)h(map,)i(in)e(others)f(b)m (y)h(pro)m(ving)g(the)g(congruence.)456 2451 y(Section)p 0 1 0 0 TeXcolorcmyk 29 w(6)p Black (#section.6) [[167 423 173 435] [1 1 1 [3 3]] [0 0 1]] pdfm 29 w(con)m(tains)30 b(a)f(table)g(with)h(man)m(y)g(sequences)i(from)d (the)g(Encyclop)s(edia)456 2568 y(in)24 b Fk(E)9 b(R)p Fs(;)27 b(in)e(particular)f(all)h(sequences)i(realized)f(b)m(y)f (oligomorphic)g(p)s(erm)m(utation)456 2684 y(groups)32 b(from)h([)p 0 1 0 0 TeXcolorcmyk(2)p Black (#cite.MR1750744) [[195 395 201 407] [1 1 1 [3 3]] [0 0 1]] pdfm (])f (that)h(fall)f(in)h Fk(E)9 b(R)32 b Fs(are)h(listed.)p Black 456 2856 a Fo(Example)38 b(5.1.)p Black Black 190 w Fs(\(1\))p Black 42 w(Man)m(y)30 b(trivial)f(sequences)k(are)c(in)g Fk(E)9 b(R)p Fs(,)30 b(among)f(them)p 0 1 0 0 TeXcolorcmyk 770 2972 a(A00004)p Black [[164 360 203 372] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=00004) pdfm (,)p 0 1 0 0 TeXcolorcmyk 32 w(A00012)p Black [[210 360 248 372] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=00012) pdfm (,)p 0 1 0 0 TeXcolorcmyk 31 w(A00079)p Black [[255 360 293 372] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=00079) pdfm 32 w(\(shifted)k(b)m(y)h(one\),)p 0 1 0 0 TeXcolorcmyk 33 w(A00203)p Black [[385 360 423 372] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000203) pdfm (.)p 0 1 0 0 TeXcolorcmyk Black 604 3088 a(\(2\))p 0 1 0 0 TeXcolorcmyk 41 w(A023890)p Black [[127 357 208 358] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=023890) pdfm (,)g(the)h(sum)h(of)e(non-prime)h(divisors,)i(is)e(in)g Fk(E)9 b(R)35 b Fs(since)h(it)f(cor-)770 3204 y(resp)s(onds)f(to)e(ha)m (ving)i(one)e(orbit)h(of)f(eac)m(h)h(comp)s(osite)h(length.)p 0 1 0 0 TeXcolorcmyk Black 604 3321 a(\(3\))p 0 1 0 0 TeXcolorcmyk 41 w(A000984)p Black [[127 330 208 330] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000984) pdfm 42 w(\(shifted)44 b(b)m(y)g(one\).)74 b(As)44 b(p)s(oin)m(ted)f(out)g (in)g(Example)p 0 1 0 0 TeXcolorcmyk 45 w(2.2)p Black (#theorem.2.2) [[458 318 473 330] [1 1 1 [3 3]] [0 0 1]] pdfm (.4,)770 3441 y(the)32 b(sequence)h(of)e(cen)m(tral)h(binomial)g(co)s(e\016cien) m(ts)2662 3360 y Fr(\000)2707 3397 y Fi(2)p Ft(n)2725 3475 y(n)2786 3360 y Fr(\001)2862 3441 y Fs(is)g(in)f Fk(E)9 b(R)31 b Fs(for)g(a)770 3574 y(com)m(binatorial)j(reason.)47 b(Similarly)35 b(the)f(sequences)i(of)d(the)h(form)3272 3494 y Fr(\000)3317 3530 y Ft(k)r(n)3320 3609 y(j)t(n)3399 3494 y Fr(\001)770 3705 y Fs(are)f(all)f(in)h Fk(E)9 b(R)p Fs(:)43 b(these)34 b(include)p 0 1 0 0 TeXcolorcmyk 34 w(A005809)p Black [[310 272 354 284] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=005809) pdfm 32 w(\()p Fp(k)c Fs(=)e(3)p Fp(;)17 b(j)33 b Fs(=)27 b(1\).)p 0 1 0 0 TeXcolorcmyk Black 604 3821 a(\(4\))p 0 1 0 0 TeXcolorcmyk 41 w(A001035)p Black [[127 270 208 270] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001035) pdfm 33 w(\(shifted)34 b(b)m(y)h(one\))f(coun)m(ts)h(the)f(n)m(um)m(b)s(er)h (of)e(distinct)i(p)s(osets)770 3937 y(on)h Fp(n)f Fs(lab)s(eled)i (elemen)m(ts.)55 b(The)36 b(\014rst)g(16)f(terms)i(of)e(this)h (sequence)i(are)770 4053 y(kno)m(wn,)k(and)c(so)h(the)g(congruence)h (\()p 0 1 0 0 TeXcolorcmyk(3)p Black (#equation.3) [[331 231 337 243] [1 1 1 [3 3]] [0 0 1]] pdfm (\))e (can)h(b)s(e)g(v)m(eri\014ed)h(for)e Fp(n)g Fk(\024)h Fs(16.)770 4169 y(Ho)m(w)m(ev)m(er,)i(the)d(sequence)i(is)e(not)f(in)h Fk(E)9 b(R)p Fs(.)57 b(W)-8 b(e)38 b(are)g(grateful)f(to)g(Greg)770 4286 y(Kup)s(erb)s(erg)23 b(for)f(suggesting)h(the)g(follo)m(wing)g (explanation.)41 b(W)-8 b(rite)23 b Fk(P)8 b Fs(\()p Fp(n)p Fs(\))770 4402 y(for)23 b(the)g(set)h(of)e(p)s(oset)i (structures)g(on)f Fh(Z)p Fp(=n)p Fh(Z)p Fs(.)35 b(Then)24 b(for)f Fp(d)p Fk(j)p Fp(n)p Fs(,)h(there)g(is)f(an)770 4518 y(injection)36 b Fp(\036)1229 4533 y Ft(d;n)1363 4518 y Fs(:)c Fk(P)8 b Fs(\()p Fp(d)p Fs(\))32 b Fk(!)g(P)8 b Fs(\()p Fp(n)p Fs(\))35 b(obtained)h(b)m(y)g(pulling)f(bac)m(k)h(a)f (p)s(oset)770 4634 y(structure)41 b(using)g(the)f(canonical)h (homomorphism)g Fh(Z)p Fp(=n)p Fh(Z)35 b Fk(!)40 b Fh(Z)p Fp(=d)p Fh(Z)p Fs(.)770 4751 y(F)-8 b(or)24 b(certain)i(v)-5 b(alues)26 b(of)f Fp(n)p Fs(,)h(including)h(all)d(prime)i(v)-5 b(alues,)28 b(w)m(e)e(claim)g(that)770 4867 y(those)42 b(p)s(osets)f(that)g(do)f(not)h(app)s(ear)g(in)f(the)i(image)f(of)f (one)h(of)f(these)770 4983 y(injections)e(come)f(in)g(families)h(of)e (size)i(a)e(m)m(ultiple)i(of)e Fp(n)p Fs(,)i(whic)m(h)g(giv)m(es)770 5099 y(the)d(congruence)g(\()p 0 1 0 0 TeXcolorcmyk(3)p Black (#equation.3) [[250 105 256 117] [1 1 1 [3 3]] [0 0 1]] pdfm (\).)48 b(b)m(y)35 b(M\177)-49 b(obius)36 b(in)m(v)m(ersion.)50 b(T)-8 b(ranslation)35 b(giv)m(es)g(an)770 5216 y(action)26 b(of)f Fh(Z)p Fp(=n)p Fh(Z)c Fs(on)26 b Fk(P)8 b Fs(\()p Fp(n)p Fs(\);)28 b(if)e(a)g(giv)m(en)h(p)s(oset)f(lies)h(on)e(a)h(free) g(orbit)g(then)p Black Black eop %%Page: 14 14 14 13 bop Black 0 TeXcolorgray Black 456 251 a Fm(14)p Black 770 450 a Fs(that)34 b(orbit)f(is)h(the)h(family)-8 b(.)47 b(In)34 b(general,)g(supp)s(ose)h(that)f(the)g(stabilizer)770 566 y(of)28 b(an)g(orbit)g(is)g Fh(Z)p Fp(=)p Fs(\()p Fp(n=d)p Fs(\))p Fh(Z)p Fs(,)23 b(but)29 b(it)f(is)g(not)g(in)g(the)h (image)f(of)g Fp(\036)3051 581 y Ft(d;n)3153 566 y Fs(.)42 b(Then)770 683 y(there)47 b(is)g(a)f(natural)g(action)g(of)f(the)i (wreath)g(pro)s(duct)f Fh(Z)p Fp(=d)p Fh(Z)p Fs(wr)-5 b Fp(S)3326 698 y Ft(n=d)770 799 y Fs(de\014ned)32 b(b)m(y)f(p)s(erm)m (uting)g(the)g(p)s(oin)m(ts)g(in)f(eac)m(h)h(coset)g(of)f Fh(Z)p Fp(=)p Fs(\()p Fp(n=d)p Fs(\))p Fh(Z)24 b Fs(and)770 915 y(adding)j(a)f(m)m(ultiple)i(of)e Fp(n=d)p Fs(.)42 b(If)26 b Fp(n)h Fs(is)g(the)g(pro)s(duct)g(of)f(t)m(w)m(o)h(primes)h (\(and)770 1031 y(for)34 b(man)m(y)h(other)f Fp(n)p Fs(\))h(then)f(the) h(size)h(of)d(the)i(orbits)f(of)g(this)h(action)f(are)770 1147 y(divisible)k(b)m(y)f Fp(n)p Fs(.)54 b(Ho)m(w)m(ev)m(er,)40 b(at)35 b Fp(n)f Fs(=)g(18)h(there)i(are)f(orbits)g(of)g(size)h Fk(\006)p Fs(6)770 1264 y(mo)s(d)c(18,)f(so)h(here)g(w)m(e)h(exp)s(ect) f(the)g(congruence)h(\()p 0 1 0 0 TeXcolorcmyk(3)p Black (#equation.3) [[389 565 395 577] [1 1 1 [3 3]] [0 0 1]] pdfm (\))f(to) f(fail.)p 0 1 0 0 TeXcolorcmyk Black 604 1380 a(\(5\))p 0 1 0 0 TeXcolorcmyk 41 w(A001945)p Black [[127 562 208 563] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001945) pdfm (:)50 b(1)p Fp(;)17 b Fs(1)p Fp(;)g Fs(1)p Fp(;)g Fs(5)p Fp(;)g Fs(1)p Fp(;)g Fs(7)p Fp(;)g Fs(8)p Fp(;)g Fs(5)p Fp(;)g Fs(19)p Fp(;)g(:)g(:)g(:)47 b Fs(is)37 b(in)g Fk(E)9 b Fp(R)36 b Fs(since)i(it)e(coun)m(ts)i(the)770 1496 y(p)s(erio)s(dic)27 b(p)s(oin)m(ts)h(in)f(the)h(automorphism)g(of) e(the)h(3-torus)g(giv)m(en)h(b)m(y)g(the)770 1728 y(matrix)1095 1528 y Fr(2)1095 1707 y(4)1161 1610 y Fs(0)83 b(1)g(0)1161 1726 y(0)g(0)g(1)1161 1843 y(1)g(1)g(0)1473 1528 y Fr(3)1473 1707 y(5)1540 1728 y Fs(.)66 b(This)41 b(sequence)i(has)d(b)s(een)h (studied)g(computa-)770 1966 y(tionally)36 b(for)g(prime)h(app)s (earances)g(\(see)f([)p 0 1 0 0 TeXcolorcmyk(4)p Black (#cite.MR1783409) [[353 481 359 493] [1 1 1 [3 3]] [0 0 1]] pdfm (]\)) g(and)g(it)g(comes)h(from)f(the)770 2082 y(cubic)e(p)s(olynomial)f (with)g(smallest)h(Mahler)f(measure)h(\(see)g([)p 0 1 0 0 TeXcolorcmyk(6)p Black (#cite.MR2000e:11087) [[439 467 445 479] [1 1 1 [3 3]] [0 0 1]] pdfm (]\).)p Black 604 2198 a(\(6\))p Black 41 w(The)48 b(large)e(class)i (of)e(elliptic)i(divisibilit)m(y)h(sequences)h(\(see)e([)p 0 1 0 0 TeXcolorcmyk(5)p Black (#cite.elliptic) [[447 453 453 465] [1 1 1 [3 3]] [0 0 1]] pdfm (]\))e (and)770 2314 y(Somos)33 b(sequences)j(seem)e(nev)m(er)g(to)e(fall)h (in)f Fk(E)9 b(R)p Fs(.)p Black 604 2430 a(\(7\))p Black 41 w(Three)26 b(in)m(teresting)h(sequences)h(that)c(seem)j(to)d(b)s(e)h (in)h Fk(E)9 b(R)24 b Fs(are)h(the)g(Euler)770 2547 y(sequence)p 0 1 0 0 TeXcolorcmyk 45 w(A000364)p Black [[214 411 258 423] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000364) pdfm 41 w(and)42 b(the)g(sequences)p 0 1 0 0 TeXcolorcmyk 46 w(A006953)p Black [[363 411 407 423] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=006953) pdfm (,)p 0 1 0 0 TeXcolorcmyk 43 w(A006954)p Black [[415 411 459 423] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=006954) pdfm 41 w(con-)770 2663 y(nected)34 b(with)f(the)g(Bernoulli)h(n)m(um)m(b)s (ers.)p Black 456 2896 a Fo(Example)50 b(5.2.)p Black 47 w Fs(Sequences)45 b(in)e Fk(E)9 b(R)43 b Fs(arise)g(from)g(the)g (com)m(binatorics)h(of)e(an)456 3012 y(iterated)30 b(map.)43 b(It)30 b(is)h(a)f(natural)g(question)h(to)f(ask)h(what)g(an)f(orbit)g (of)f Fo(ORBIT)456 3129 y Fs(lo)s(oks)47 b(lik)m(e,)52 b(and)47 b(whether)h(there)g(are)f(an)m(y)g(asymptotic)h(prop)s(erties) g(asso)s(ci-)456 3245 y(ated)35 b(to)g(it.)51 b(The)36 b(simplest)h(orbit)e(starts)g(with)h(the)f(unit)h(sequence)p 0 1 0 0 TeXcolorcmyk 38 w(A000007)p Black [[438 328 482 340] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000007) pdfm (.)456 3361 y(Applying)43 b Fo(ORBIT)f Fs(iterativ)m(ely)j(to)d(this)i (giv)m(es)g(the)f(follo)m(wing)g(sequence)j(of)456 3477 y(sequences)e(\(in)d(eac)m(h)h(case,)i(the)e(sequence)h(coun)m(ts)g (the)e(n)m(um)m(b)s(er)h(of)f(p)s(erio)s(dic)456 3594 y(p)s(oin)m(ts)c(in)g(a)g(map)g(whic)m(h)i(has)e(the)h(n)m(um)m(b)s(er) g(of)f(orbits)g(of)g(length)g Fp(n)g Fs(giv)m(en)h(b)m(y)456 3710 y(the)33 b Fp(n)p Fs(th)f(en)m(try)i(in)f(the)g(previous)h (sequence\).)456 3847 y(1)p Fp(;)17 b Fs(0)p Fp(;)g Fs(0)p Fp(;)g Fs(0)p Fp(;)g Fs(0)p Fp(;)g Fs(0)p Fp(;)g Fs(0)p Fp(;)g Fs(0)p Fp(;)g Fs(0)p Fp(;)g Fs(0)p Fp(;)g Fs(0)p Fp(;)g Fs(0)p Fp(;)f Fs(0)p Fp(;)h Fs(0)p Fp(;)f Fs(0)p Fp(;)h Fs(0)p Fp(;)f Fs(0)p Fp(;)h Fs(0)o Fp(;)g Fs(0)p Fp(;)g Fs(0)o Fp(;)g Fs(0)p Fp(;)g Fs(0)o Fp(;)g Fs(0)p Fp(;)g Fs(0)o Fp(;)g Fs(0)p Fp(;)g Fs(0)o Fp(;)121 b Fs(\()p 0 1 0 0 TeXcolorcmyk Fp(A)p Fs(000007)p [[433 255 477 267] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000007) pdfm Black [[433 255 477 267] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000007) pdfm -1 w(\))456 3984 y(1)p Fp(;)17 b Fs(1)p Fp(;)g Fs(1)p Fp(;)g Fs(1)p Fp(;)g Fs(1)p Fp(;)g Fs(1)p Fp(;)g Fs(1)p Fp(;)g Fs(1)p Fp(;)g Fs(1)p Fp(;)g Fs(1)p Fp(;)g Fs(1)p Fp(;)g Fs(1)p Fp(;)f Fs(1)p Fp(;)h Fs(1)p Fp(;)f Fs(1)p Fp(;)h Fs(1)p Fp(;)f Fs(1)p Fp(;)h Fs(1)o Fp(;)g Fs(1)p Fp(;)g Fs(1)o Fp(;)g Fs(1)p Fp(;)g Fs(1)o Fp(;)g Fs(1)p Fp(;)g Fs(1)o Fp(;)g Fs(1)p Fp(;)g Fs(1)147 b(\()p 0 1 0 0 TeXcolorcmyk Fp(A)p Fs(000012)p [[433 239 477 251] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000012) pdfm Black [[433 239 477 251] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000012) pdfm -1 w(\))456 4120 y(1)p Fp(;)17 b Fs(3)p Fp(;)g Fs(4)p Fp(;)g Fs(7)p Fp(;)g Fs(6)p Fp(;)g Fs(12)p Fp(;)g Fs(8)p Fp(;)g Fs(15)p Fp(;)g Fs(13)p Fp(;)g Fs(18)p Fp(;)f Fs(12)p Fp(;)h Fs(2)o(8)p Fp(;)g Fs(14)p Fp(;)f Fs(24)p Fp(;)g Fs(24)p Fp(;)h Fs(3)o(1)p Fp(;)g Fs(18)p Fp(;)f Fs(39)p Fp(;)g Fs(20)161 b(\()p 0 1 0 0 TeXcolorcmyk Fp(A)p Fs(000203)p [[433 223 477 235] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000203) pdfm Black [[433 223 477 235] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000203) pdfm -1 w(\))456 4257 y(1)p Fp(;)17 b Fs(7)p Fp(;)g Fs(13)p Fp(;)g Fs(35)p Fp(;)g Fs(31)p Fp(;)g Fs(91)p Fp(;)g Fs(57)p Fp(;)g Fs(155)p Fp(;)g Fs(13)o(0)p Fp(;)g Fs(21)o(7)p Fp(;)g Fs(13)o(3)p Fp(;)g Fs(45)o(5)p Fp(;)g Fs(18)o(3)p Fp(;)g Fs(399)o Fp(;)g Fs(403)140 b(\()p 0 1 0 0 TeXcolorcmyk Fp(A)p Fs(001001)p [[433 206 477 218] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001001) pdfm Black [[433 206 477 218] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001001) pdfm -1 w(\))456 4394 y(1)p Fp(;)17 b Fs(15)p Fp(;)g Fs(40)p Fp(;)g Fs(155)p Fp(;)g Fs(156)p Fp(;)g Fs(600)p Fp(;)g Fs(400)p Fp(;)g Fs(1)o(395)p Fp(;)f Fs(1210)p Fp(;)g Fs(2340)o Fp(;)h Fs(146)o(4)p Fp(;)g Fs(62)o(00)p Fp(;)g Fs(2)o(380)p Fp(;)f Fs(6000)456 4531 y(1)p Fp(;)h Fs(31)p Fp(;)g Fs(121)p Fp(;)g Fs(651)p Fp(;)g Fs(781)p Fp(;)g Fs(3751)p Fp(;)g Fs(2801)o Fp(;)g Fs(118)o(11)p Fp(;)g Fs(1)o(1011)o Fp(;)g Fs(242)o(11)p Fp(;)g Fs(16)o(105)p Fp(;)f Fs(7877)o(1)p Fp(;)h Fs(30)o(941)456 4668 y(1)p Fp(;)g Fs(63)p Fp(;)g Fs(364)p Fp(;)g Fs(2667)p Fp(;)g Fs(3906)p Fp(;)g Fs(22932)p Fp(;)g Fs(1)o(9608)o Fp(;)g Fs(971)o(55)p Fp(;)g Fs(9)o(9463)p Fp(;)f Fs(2460)o(78)p Fp(;)h Fs(1)o(7715)o(6)456 4805 y(1)p Fp(;)g Fs(127)p Fp(;)g Fs(1093)p Fp(;)g Fs(10795)p Fp(;)g Fs(19531)p Fp(;)g Fs(138)o(811)p Fp(;)f Fs(13725)o(7)p Fp(;)h Fs(78)o(8035)o Fp(;)g Fs(896)o(260)p Fp(;)f Fs(24804)o(37)456 4942 y(1)p Fp(;)h Fs(255)p Fp(;)g Fs(3280)p Fp(;)g Fs(43435)p Fp(;)g Fs(97656)p Fp(;)g Fs(836)o(400)p Fp(;)f Fs(96080)o(0)p Fp(;)h Fs(63)o(4771)o(5)p Fp(;)g Fs(80)o(69620)o Fp(;)g Fs(249)o(0228)o(0)456 5079 y(1)p Fp(;)g Fs(511)p Fp(;)g Fs(9841)p Fp(;)g Fs(174251)p Fp(;)g Fs(488281)p Fp(;)g Fs(5)o(0287)o(51)p Fp(;)g Fs(67)o(2560)o(1)p Fp(;)g Fs(50)o(95597)o(1)p Fp(;)g Fs(72)o(6364)o(21)456 5216 y(1)p Fp(;)g Fs(1023)p Fp(;)g Fs(29524)p Fp(;)g Fs(698027)p Fp(;)g Fs(244140)o(6)p Fp(;)g Fs(30)o(20305)o(2)p Fp(;)g Fs(47)o(0792)o(08)p Fp(;)g Fs(40)o(8345)o(795)p Black Black eop %%Page: 15 15 15 14 bop Black 0 TeXcolorgray Black 3368 251 a Fm(15)p Black 456 450 a Fs(1)p Fp(;)17 b Fs(2047)p Fp(;)g Fs(88573)p Fp(;)g Fs(2794155)p Fp(;)g Fs(12207)o(031)p Fp(;)f Fs(18130)o(8931)o Fp(;)h Fs(329)o(55445)o(7)p Fp(;)g Fs(32)o(6956)o(0515)456 590 y(1)p Fp(;)g Fs(4095)p Fp(;)g Fs(265720)p Fp(;)g Fs(11180715)p Fp(;)g Fs(610)o(35156)o Fp(;)g Fs(108)o(8123)o(400)p Fp(;)f Fs(23068)o(8120)o(0)456 730 y(1)p Fp(;)h Fs(8191)p Fp(;)g Fs(797161)p Fp(;)g Fs(44731051)p Fp(;)g Fs(305)o(17578)o(1)p Fp(;)g Fs(65)o(2954)o(5751)p Fp(;)f Fs(1614)o(8168)o(401)456 870 y(1)p Fp(;)h Fs(16383)p Fp(;)g Fs(2391484)p Fp(;)g Fs(178940587)p Fp(;)f Fs(15258)o(7890)o(6)p Fp(;)h Fs(39)o(17968)o (2372)o Fp(;)g Fs(113)o(03717)o(8808)456 1010 y(1)p Fp(;)g Fs(32767)p Fp(;)g Fs(7174453)p Fp(;)g Fs(715795115)p Fp(;)f Fs(76293)o(9453)o(1)p Fp(;)h Fs(23)o(50853)o(0145)o(1)456 1150 y(1)p Fp(;)g Fs(65535)p Fp(;)g Fs(21523360)p Fp(;)g Fs(2863245995)o Fp(;)g Fs(381)o(4697)o(2656)p Fp(;)f Fs(1410)o(5333)o(97600)555 1290 y(The)33 b(arithmetic)g(and)f(gro)m (wth)g(prop)s(erties)h(of)e(these)i(sequences)i(will)e(b)s(e)f(ex-)456 1406 y(plored)37 b(elsewhere.)58 b(The)37 b(sequence)i(of)d(\014rst,)j (second)e(and)g(third)g(terms)g(com-)456 1522 y(prise)p 0 1 0 0 TeXcolorcmyk 33 w(A000012)p Black [[155 534 199 546] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000012) pdfm (,)p 0 1 0 0 TeXcolorcmyk 32 w(A000225)p Black [[206 534 250 546] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000225) pdfm (,)31 b(and)p 0 1 0 0 TeXcolorcmyk 33 w(A003462)p Black [[280 534 324 546] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=003462) pdfm 31 w(resp)s(ectiv)m(ely)-8 b(.)1665 1739 y(6.)48 b Fl(Summar)-7 b(y)555 1914 y Fs(Being)31 b(exactly)g(realizable)h(is)e(a)g(strong)g (symmetry)j(prop)s(ert)m(y)e(of)e(an)h(in)m(teger)456 2030 y(sequence.)45 b(In)31 b(this)h(table)f(w)m(e)g(summarize)i(the)e (sequences)j(from)c(the)h(Encyclo-)456 2146 y(p)s(edia)36 b(found)h(to)f(b)s(e)g(exactly)i(realizable,)g(together)f(with)g(the)f (corresp)s(onding)456 2262 y(sequence)41 b(coun)m(ting)f(the)f(orbits,) i(and)e(an)m(y)g(other)g(information.)63 b(All)39 b(the)g(se-)456 2379 y(quences)f(are)d(exp)s(ected)j(to)d(ha)m(v)m(e)i(realizing)f (maps)g({)g(the)g(inclusion)h(of)e(a)g(map)456 2495 y(means)40 b(that)g(w)m(e)g(kno)m(w)h(of)e(a)g(map)h(that)f(is)h(natural)g(in)f (some)i(sense)g(\(for)e(ex-)456 2611 y(ample,)45 b(has)e(a)f(\014nite)h (description)h(or)e(is)h(algebraic\).)73 b(Direct)42 b(pro)s(ofs)g(of)g(the)456 2727 y(congruence)33 b(are)e(cited)i(in)f (some)g(brief)g(fashion)g({)f(a)h(question)h(mark)f(indicates)456 2843 y(that)37 b(w)m(e)i(do)f(not)g(kno)m(w)h(a)e(pro)s(of)g(and)h (seek)i(one,)f(e)f(means)h(it)f(is)g(easy)-8 b(,)41 b(and)d(a)456 2960 y(com)m(binatorial)28 b(coun)m(ting)g(problem)h(suggests)g(wh)m(y) g(the)f(n)m(um)m(b)s(er)h(of)e(orbits)h(is)h(a)456 3076 y(non-negativ)m(e)34 b(in)m(teger.)48 b(The)34 b(com)m(binatorial)h (coun)m(ting)f(problems)h(and)f(maps)456 3192 y(are)e(lab)s(elled)i(as) e(follo)m(ws.)p Black 679 3332 a Fk(\017)p Black 41 w Fs(POL)-8 b(Y:)34 b(the)g(orbits)f(coun)m(t)i(the)e(n)m(um)m(b)s(er)i (of)e(irreducible)i(p)s(olynomials)770 3448 y(o)m(v)m(er)f(a)e (\014nite)h(\014eld.)p Black 679 3565 a Fk(\017)p Black 41 w Fs(NECK\()p Fp(k)s Fs(\):)50 b(the)36 b(orbits)g(coun)m(t)h(the)f (n)m(um)m(b)s(er)h(of)e(ap)s(erio)s(dic)h(nec)m(klaces)770 3681 y(with)d Fp(n)g Fs(b)s(eads)g(in)g Fp(k)j Fs(colours.)p Black 679 3797 a Fk(\017)p Black 41 w Fs(NECK:)31 b(the)f(orbits)g (coun)m(t)h(a)e(family)i(of)e(nec)m(klaces)j(with)f(constrain)m(t)g({) 770 3913 y(see)j(the)f(encyclop)s(edia)h(en)m(try)g(for)e(details.)p Black 679 4030 a Fk(\017)p Black 41 w Fs(KUMMER:)j(follo)m(ws)g(from)e (the)h(Kummer)h(and)f(v)m(on)h(Staudt)f(congru-)770 4146 y(ences.)p Black 679 4262 a Fk(\017)p Black 41 w Fs(COMB:)g(follo)m(ws) f(from)f(standard)h(com)m(binatorics)h(argumen)m(ts.)p Black 679 4378 a Fk(\017)p Black 41 w Fs(CHK:)f(the)g(orbit)g(sequence) i(is)e(a)f(`CHK')h(transform.)p Black 679 4494 a Fk(\017)p Black 41 w Fs(S\(1\):)p Fp(S)6 b Fs(-in)m(teger)33 b(map)g(with)g Fp(\030)f Fs(=)27 b(2)p Fp(;)17 b(S)33 b Fs(=)28 b Fk(f)p Fs(2)p Fp(;)17 b Fs(3)p Fk(g)p Fp(;)g(k)29 b Fs(=)f Fh(Q)11 b Fs(.)p Black 679 4611 a Fk(\017)p Black 41 w Fs(S\(2\):)p Fp(S)6 b Fs(-in)m(teger)33 b(map)g(with)g Fp(\030)f Fs(=)27 b Fp(t;)17 b(S)34 b Fs(=)27 b Fk(f)p Fp(t)c Fs(+)f(1)p Fk(g)p Fp(;)17 b(k)30 b Fs(=)d Fh(F)2878 4626 y Fi(2)2924 4611 y Fs(\()p Fp(t)p Fs(\).)p Black 679 4727 a Fk(\017)p Black 41 w Fs(R:)33 b(irrational)f(circle)h(rotation.)456 4867 y(Of)d(course)h(there)g(are)g(often)f(man)m(y)i(w)m(a)m(ys)g(to)e (\014ll)g(in)h(the)g(last)g(column.)43 b(If)31 b(there)456 4983 y(is)39 b(a)f(natural)h(realizing)g(map,)i(then)e(that)g(fact)f (in)h(itself)g(is)h(usually)g(the)f(b)s(est)456 5099 y(pro)s(of)23 b(of)h(the)h(congruence.)42 b(Sequences)27 b(mark)m(ed)f(with)f(a)f(question)i(mark)e(in)h(the)456 5216 y(\014rst)i(column)h(are)f(not)h(kno)m(wn)g(to)f(b)s(e)g(in)h Fk(E)9 b(R)27 b Fs(at)g(all:)40 b(they)28 b(just)g(seem)h(to)d(satisfy) p Black Black eop %%Page: 16 16 16 15 bop Black 0 TeXcolorgray Black 456 251 a Fm(16)p Black 456 450 a Fs(the)36 b(congruence)h(for)f(the)g(\014rst)g(t)m(w)m (en)m(t)m(y)i(or)e(so)g(terms.)54 b(A)36 b(star)g(indicates)i(that)456 566 y(the)e(initial)f(term)h(of)f(the)h(sequence)i(is)e(shifted)h(b)m (y)f(one.)53 b(Of)35 b(course)i(an)m(y)f(non-)456 683 y(negativ)m(e)25 b(in)m(teger)g(sequence)i(at)d(all)g(can)g(app)s(ear)g (in)h(the)f(second)i(column,)h(so)d(the)456 799 y(selection)38 b(here)f(is)g(based)g(on)f(the)h(follo)m(wing)g(arbitrary)f(criterion:) 52 b(either)37 b(the)456 915 y(p)s(erio)s(dic)c(p)s(oin)m(t)f(sequence) k(or)c(the)h(orbit)f(coun)m(ting)i(sequence)h(is)e(`in)m(teresting'.)p 562 1019 2776 4 v 562 1019 V 604 1100 a Fp(f)652 1115 y Ft(n)699 1100 y Fs(\()p Fp(T)14 b Fs(\))578 b Fp(f)1483 1064 y Ft(o)1472 1125 y(n)1521 1100 y Fs(\()p Fp(T)14 b Fs(\))253 b Fp(T)806 b Fs(Pro)s(of)32 b(of)g(\()p 0 1 0 0 TeXcolorcmyk(3)p Black (#equation.3) [[456 585 462 597] [1 1 1 [3 3]] [0 0 1]] pdfm (\))p 562 1139 V 562 1139 V 0 1 0 0 TeXcolorcmyk 604 1336 a(A00004)p Black [[144 557 183 569] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=00004) pdfm 0 1 0 0 TeXcolorcmyk 502 w(A00004)p Black [[243 557 281 569] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=00004) pdfm 179 w(R)791 b(e)p 0 1 0 0 TeXcolorcmyk 604 1452 a(A00012)p Black [[144 543 183 555] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=00012) pdfm 0 1 0 0 TeXcolorcmyk 502 w(A00007)p Black [[243 543 281 555] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=00007) pdfm 179 w(singleton)484 b(e)p 0 1 0 0 TeXcolorcmyk 604 1569 a(A000079*)p Black [[144 529 194 541] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000079) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A001037)p Black [[243 529 287 541] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001037) pdfm 130 w(full)33 b(2-shift)423 b(POL)-8 b(Y)p 0 1 0 0 TeXcolorcmyk 604 1685 a(A000203)p Black [[144 515 189 527] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000203) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A00012)p Black [[243 515 281 527] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=00012) pdfm 179 w(-)830 b(e)p 0 1 0 0 TeXcolorcmyk 604 1801 a(A000204)p Black [[144 501 189 513] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000204) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A006206)p Black [[243 501 287 513] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=006206) pdfm 130 w(golden)33 b(mean)g(shift)107 b(NECK)p 0 1 0 0 TeXcolorcmyk 604 1917 a(A000244*)p Black [[144 487 194 499] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000244) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A027376*)p Black [[243 487 293 499] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=027376) pdfm 81 w(full)33 b(3-shift)423 b(POL)-8 b(Y)p 0 1 0 0 TeXcolorcmyk 604 2034 a(A000302*)p Black [[144 473 194 485] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000302) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A027377*)p Black [[243 473 293 485] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=027377) pdfm 81 w(full)33 b(4-shift)423 b(POL)-8 b(Y)p 0 1 0 0 TeXcolorcmyk 604 2150 a(A000351*)p Black [[144 459 194 471] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000351) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A001692*)p Black [[243 459 293 471] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001692) pdfm 81 w(full)33 b(5-shift)423 b(POL)-8 b(Y)p 0 1 0 0 TeXcolorcmyk 604 2266 a(A000364*?)p Black [[144 445 200 457] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000364) pdfm 0 1 0 0 TeXcolorcmyk 358 w(A060164)p Black [[243 445 287 457] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060164) pdfm 130 w(-)830 b(-)p 0 1 0 0 TeXcolorcmyk 604 2382 a(A000400*)p Black [[144 431 194 443] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000400) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A032164)p Black [[243 431 287 443] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=032164) pdfm 130 w(full)33 b(6-shift)423 b(NECK\(6\))p 0 1 0 0 TeXcolorcmyk 604 2499 a(A000420*)p Black [[144 417 194 429] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000420) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A001693)p Black [[243 417 287 429] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001693) pdfm 130 w(full)33 b(7-shift)423 b(POL)-8 b(Y)p 0 1 0 0 TeXcolorcmyk 604 2615 a(A000593)p Black [[144 403 189 415] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000593) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A000035*)p Black [[243 403 293 415] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000035) pdfm 81 w(-)830 b(e)p 0 1 0 0 TeXcolorcmyk 604 2731 a(A000670*)p Black [[144 389 194 401] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000670) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A060223)p Black [[243 389 287 401] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060223) pdfm 130 w(e)p 0 1 0 0 TeXcolorcmyk 604 2847 a(A000984*)p Black [[144 375 194 387] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000984) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A060165)p Black [[243 375 287 387] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060165) pdfm 130 w(-)g(COMB)p 0 1 0 0 TeXcolorcmyk 604 2963 a(A001001)p Black [[144 361 189 373] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001001) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A000203)p Black [[243 361 287 373] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000203) pdfm 130 w(-)g(e)p 0 1 0 0 TeXcolorcmyk 604 3080 a(A001018*)p Black [[144 347 194 359] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001018) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A027380*)p Black [[243 347 293 359] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=027380) pdfm 81 w(full)33 b(8-shift)423 b(POL)-8 b(Y)p 0 1 0 0 TeXcolorcmyk 604 3196 a(A001019*)p Black [[144 333 194 345] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001019) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A027381*)p Black [[243 333 293 345] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=027381) pdfm 81 w(full)33 b(9-shift)423 b(POL)-8 b(Y)p 0 1 0 0 TeXcolorcmyk 604 3312 a(A001020*)p Black [[144 320 194 332] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001020) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A032166)p Black [[243 320 287 332] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=032166) pdfm 130 w(full)33 b(11-shift)374 b(NECK\(11\))p 0 1 0 0 TeXcolorcmyk 604 3428 a(A001021*)p Black [[144 306 194 318] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001021) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A032167)p Black [[243 306 287 318] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=032167) pdfm 130 w(full)33 b(12-shift)374 b(NECK\(12\))p 0 1 0 0 TeXcolorcmyk 604 3545 a(A001022*)p Black [[144 292 194 304] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001022) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A060216)p Black [[243 292 287 304] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060216) pdfm 130 w(full)33 b(13-shift)374 b(NECK\(13\))p 0 1 0 0 TeXcolorcmyk 604 3661 a(A001023*)p Black [[144 278 194 290] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001023) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A060217)p Black [[243 278 287 290] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060217) pdfm 130 w(full)33 b(14-shift)374 b(NECK\(14\))p 0 1 0 0 TeXcolorcmyk 604 3777 a(A001024*)p Black [[144 264 194 276] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001024) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A060218)p Black [[243 264 287 276] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060218) pdfm 130 w(full)33 b(15-shift)374 b(NECK\(15\))p 0 1 0 0 TeXcolorcmyk 604 3893 a(A001025*)p Black [[144 250 194 262] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001025) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A060219)p Black [[243 250 287 262] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060219) pdfm 130 w(full)33 b(16-shift)374 b(NECK\(16\))p 0 1 0 0 TeXcolorcmyk 604 4010 a(A001026*)p Black [[144 236 194 248] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001026) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A060220)p Black [[243 236 287 248] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060220) pdfm 130 w(full)33 b(17-shift)374 b(NECK\(17\))p 0 1 0 0 TeXcolorcmyk 604 4126 a(A001027*)p Black [[144 222 194 234] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001027) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A060221)p Black [[243 222 287 234] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060221) pdfm 130 w(full)33 b(18-shift)374 b(NECK\(18\))p 0 1 0 0 TeXcolorcmyk 604 4242 a(A001029*)p Black [[144 208 194 220] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001029) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A060222)p Black [[243 208 287 220] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060222) pdfm 130 w(full)33 b(19-shift)374 b(NECK\(19\))p 0 1 0 0 TeXcolorcmyk 604 4358 a(A001157)p Black [[144 194 189 206] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001157) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A000027)p Black [[243 194 287 206] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000027) pdfm 130 w(-)830 b(e)p 0 1 0 0 TeXcolorcmyk 604 4474 a(A001158)p Black [[144 180 189 192] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001158) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A000290*)p Black [[243 180 293 192] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000290) pdfm 81 w(-)g(e)p 0 1 0 0 TeXcolorcmyk 604 4591 a(A001641?)p Black [[144 166 194 178] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001641) pdfm 0 1 0 0 TeXcolorcmyk 407 w(A060166)p Black [[243 166 287 178] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060166) pdfm 130 w(-)g(-)p 0 1 0 0 TeXcolorcmyk 604 4707 a(A001642?)p Black [[144 152 194 164] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001642) pdfm 0 1 0 0 TeXcolorcmyk 407 w(A060167)p Black [[243 152 287 164] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060167) pdfm 130 w(-)g(-)p 0 1 0 0 TeXcolorcmyk 604 4823 a(A001643?)p Black [[144 138 194 150] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001643) pdfm 0 1 0 0 TeXcolorcmyk 407 w(A060168)p Black [[243 138 287 150] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060168) pdfm 130 w(-)g(-)p 0 1 0 0 TeXcolorcmyk 604 4939 a(A001700)p Black [[144 124 189 136] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001700) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A022553)p Black [[243 124 287 136] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=022553) pdfm 130 w(-)g(-)p 0 1 0 0 TeXcolorcmyk 604 5056 a(A001945)p Black [[144 110 189 122] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001945) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A060169)p Black [[243 110 287 122] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060169) pdfm 130 w(auto)32 b(of)h Fh(T)2318 5019 y Fi(3)2784 5056 y Fs(-)p 0 1 0 0 TeXcolorcmyk 604 5172 a(A004146*)p Black [[144 96 194 108] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=004146) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A032170)p Black [[243 96 287 108] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=032170) pdfm 130 w(auto)f(of)h Fh(T)2318 5136 y Fi(2)2784 5172 y Fs(CHK)p Black Black eop %%Page: 17 17 17 16 bop Black 0 TeXcolorgray Black 3368 251 a Fm(17)p Black 0 1 0 0 TeXcolorcmyk 604 450 a Fs(A005809*)p Black [[144 663 194 675] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=005809) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A060170)p Black [[243 663 287 675] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060170) pdfm 130 w(-)830 b(COMB)p 0 1 0 0 TeXcolorcmyk 604 566 a(A006953?)p Black [[144 649 194 661] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=006953) pdfm 0 1 0 0 TeXcolorcmyk 407 w(A060171)p Black [[243 649 287 661] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060171) pdfm 130 w(-)g(KUMMER?)p 0 1 0 0 TeXcolorcmyk 604 683 a(A006954?)p Black [[144 635 194 647] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=006954) pdfm 0 1 0 0 TeXcolorcmyk 407 w(A060479)p Black [[243 635 287 647] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060479) pdfm 130 w(-)g(KUMMER?)p 0 1 0 0 TeXcolorcmyk 604 799 a(A011557*)p Black [[144 621 194 633] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=011557) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A032165*)p Black [[243 621 293 633] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=032165) pdfm 81 w(full)33 b(10-shift)374 b(NECK\(10\))p 0 1 0 0 TeXcolorcmyk 604 915 a(A023890)p Black [[144 607 189 619] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=023890) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A005171)p Black [[243 607 287 619] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=005171) pdfm 130 w(-)830 b(e)p 0 1 0 0 TeXcolorcmyk 604 1031 a(A027306*)p Black [[144 593 194 605] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=027306) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A060172)p Black [[243 593 287 605] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060172) pdfm 130 w(-)g(COMB)p 0 1 0 0 TeXcolorcmyk 604 1147 a(A035316)p Black [[144 579 189 591] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=035316) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A010052*)p Black [[243 579 293 591] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=010052) pdfm 81 w(-)g(e)p 0 1 0 0 TeXcolorcmyk 604 1264 a(A047863*)p Black [[144 565 194 577] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=047863) pdfm 0 1 0 0 TeXcolorcmyk 404 w(A060224)p Black [[243 565 287 577] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060224) pdfm 130 w(-)g(-)p 0 1 0 0 TeXcolorcmyk 604 1380 a(A048578)p Black [[144 551 189 563] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=048578) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A060477)p Black [[243 551 287 563] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060477) pdfm 130 w(4-shift)33 b Fk([)g Fs(singleton)83 b(-)p 0 1 0 0 TeXcolorcmyk 604 1496 a(A056045)p Black [[144 537 189 549] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=056045) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A060173)p Black [[243 537 287 549] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060173) pdfm 130 w(-)830 b(COMB)604 1612 y(0,2,0,6,0,8,0,14,...)p 0 1 0 0 TeXcolorcmyk 82 w(A000035)p Black [[243 524 287 536] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=000035) pdfm 130 w(-)g(e)p 0 1 0 0 TeXcolorcmyk 604 1729 a(A059928)p Black [[144 510 189 522] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=059928) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A060478)p Black [[243 510 287 522] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060478) pdfm 130 w(auto)32 b(of)h Fh(T)2318 1692 y Fi(10)2784 1729 y Fs(e)p 0 1 0 0 TeXcolorcmyk 604 1845 a(A059990)p Black [[144 496 189 508] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=059990) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A060480)p Black [[243 496 287 508] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060480) pdfm 130 w(S\(1\))684 b(-)p 0 1 0 0 TeXcolorcmyk 604 1961 a(A059991)p Black [[144 482 189 494] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=059991) pdfm 0 1 0 0 TeXcolorcmyk 453 w(A060481)p Black [[243 482 287 494] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com:80/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=060481) pdfm 130 w(S\(2\))g(-)p 562 1999 2776 4 v 562 1999 V Black 1125 2081 a(T)-8 b(able)33 b(1:)44 b(Exactly)34 b(realizable)f (sequences.)p Black 1660 2394 a Fl(References)p Black 456 2552 a Fn([1])p Black 41 w(M.)d(Bernstein)g(and)f(N.J.A.)i(Sloane.) e(Some)h(canonical)f(sequences)g(of)h(in)n(tegers,)p 0 1 0 0 TeXcolorcmyk 30 w Fc(Line)l(ar)p [[457 411 485 423] [1 1 1 [3 3]] [0 0 1]] (http://www.elsevier.com/locate/laa) pdfm 585 2652 a(A)n(lgebr)l(a)g(Appl.)p Black [[127 399 201 411] [1 1 1 [3 3]] [0 0 1]] (http://www.elsevier.com/locate/laa) pdfm 31 w Fn(,)e Fb(226/228)p Fn(:57{72,)23 b(1995.)p Black 456 2751 a([2])p Black 41 w(P)n(eter)29 b(J.)i(Cameron.)e(Sequences)h (realized)g(b)n(y)g(oligomorphic)f(p)r(erm)n(utation)h(groups,)p 0 1 0 0 TeXcolorcmyk 30 w Fc(J.)p [[477 387 485 399] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/~njas/sequences/JIS/) pdfm 585 2851 a(Inte)l(ger)f(Se)l(q.)p Black [[127 375 193 387] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/~njas/sequences/JIS/) pdfm 30 w Fn(,)f Fb(3)p Fn(:Article)f(00.1.5,)f(h)n(tml)i(do)r(cumen)n(t)g (\(electronic\),)f(2000.)p Black 456 2951 a([3])p Black 41 w(Vija)n(y)35 b(Chothi,)i(Graham)d(Ev)n(erest,)h(and)g(Thomas)g(W)-7 b(ard.)34 b Fa(S)5 b Fn(-in)n(teger)34 b(dynamical)g(sys-)585 3050 y(tems:)j(p)r(erio)r(dic)27 b(p)r(oin)n(ts.)p 0 1 0 0 TeXcolorcmyk 28 w Fc(J.)j(R)l(eine)f(A)n(ngew.)h(Math.)p Black [[241 351 341 363] [1 1 1 [3 3]] [0 0 1]] (http://www.deGruyter.de/journals/crelle/) pdfm 31 w Fn(,)e Fb(489)p Fn(:99{132,)c(1997.)p Black 456 3150 a([4])p Black 41 w(Manfred)29 b(Einsiedler,)h(Graham)f(Ev)n (erest,)g(and)h(Thomas)f(W)-7 b(ard.)29 b(Primes)g(in)h(sequences)585 3250 y(asso)r(ciated)e(to)i(p)r(olynomials)e(\(after)i(Lehmer\).)p 0 1 0 0 TeXcolorcmyk 30 w Fc(LMS)i(J.)f(Comput.)i(Math.)p Black [[329 327 432 339] [1 1 1 [3 3]] [0 0 1]] (http://www.lms.ac.uk/jcm/) pdfm 33 w Fn(,)d Fb(3)p Fn(:125{139)585 3349 y(\(electronic\),)d(2000.)p Black 456 3449 a([5])p Black 41 w(Manfred)40 b(Einsiedler,)k(Graham)c (Ev)n(erest,)i(and)f(Thomas)f(W)-7 b(ard.)41 b(Primes)f(in)h(elliptic) 585 3548 y(divisibilit)n(y)27 b(sequences.)p 0 1 0 0 TeXcolorcmyk 27 w Fc(LMS)j(J.)g(Comput.)g(Math.)p Black [[239 291 341 303] [1 1 1 [3 3]] [0 0 1]] (http://www.lms.ac.uk/jcm/) pdfm 31 w Fn(,)e Fb(4)p Fn(:1{13)e(\(electronic\),)h(2001.)p Black 456 3648 a([6])p Black 41 w(Graham)40 b(Ev)n(erest)f(and)i (Thomas)f(W)-7 b(ard,)p 0 1 0 0 TeXcolorcmyk 44 w Fc(Heights)42 b(of)h(p)l(olynomials)h(and)f(entr)l(opy)f(in)p [[313 279 485 291] [1 1 1 [3 3]] [0 0 1]] (http://www.mth.uea.ac.uk/~h720/headings/webpage.html) pdfm 585 3748 a(algebr)l(aic)32 b(dynamics)p Black [[127 267 224 279] [1 1 1 [3 3]] [0 0 1]] (http://www.mth.uea.ac.uk/~h720/headings/webpage.html) pdfm 31 w Fn(,)27 b(Springer-V)-7 b(erlag,)25 b(London,)i(1999.)p Black 456 3847 a([7])p Black 41 w(G.H.)h(Hardy)g(and)g(E.M.)g(W)-7 b(righ)n(t.)27 b Fc(A)n(n)j(intr)l(o)l(duction)g(to)g(the)g(The)l(ory)i (of)f(Numb)l(ers)p Fn(.)c(The)585 3947 y(Clarendon)f(Press)g(Oxford)h (Univ)n(ersit)n(y)g(Press,)f(New)i(Y)-7 b(ork,)26 b(\014fth)j(edition,) f(1979.)p Black 456 4047 a([8])p Black 41 w(D.A.)j(Lind.)g(Dynamical)f (prop)r(erties)f(of)i(quasih)n(yp)r(erb)r(olic)e(toral)g (automorphisms.)p 0 1 0 0 TeXcolorcmyk 29 w Fc(Er-)p [[471 231 485 243] [1 1 1 [3 3]] [0 0 1]] (http://uk.cambridge.org/journals/ets/) pdfm 585 4146 a(go)l(dic)i(The)l(ory)g(Dynamic)l(al)g(Systems)p Black [[127 219 286 231] [1 1 1 [3 3]] [0 0 1]] (http://uk.cambridge.org/journals/ets/) pdfm 29 w Fn(,)d Fb(2)p Fn(\(1\):49{68,)d(1982.)p Black 456 4246 a([9])p Black 41 w(D.A.)37 b(Lind.)g(The)g(en)n(tropies)f(of)h (top)r(ological)e(Mark)n(o)n(v)f(shifts)k(and)e(a)h(related)f(class)g (of)585 4345 y(algebraic)26 b(in)n(tegers.)p 0 1 0 0 TeXcolorcmyk 26 w Fc(Er)l(go)l(dic)31 b(The)l(ory)g(Dynamic)l(al)g (Systems)p Black [[224 196 378 208] [1 1 1 [3 3]] [0 0 1]] (http://uk.cambridge.org/journals/ets/) pdfm 29 w Fn(,)d Fb(4)p Fn(\(2\):283{300,)c(1984.)p Black 456 4445 a([10])p Black 40 w(Douglas)29 b(Lind)i(and)f(Brian)f(Marcus.) p 0 1 0 0 TeXcolorcmyk 30 w Fc(A)n(n)i(Intr)l(o)l(duction)h(to)g(Symb)l (olic)h(Dynamics)g(and)p [[295 184 485 196] [1 1 1 [3 3]] [0 0 1]] (http://www.math.washington.edu/SymbolicDynamics/) pdfm 585 4545 a(Co)l(ding)p Black [[127 172 172 184] [1 1 1 [3 3]] [0 0 1]] (http://www.math.washington.edu/SymbolicDynamics/) pdfm 31 w Fn(.)28 b(Cam)n(bridge)e(Univ)n(ersit)n(y)g(Press,)h(Cam)n (bridge,)f(1995.)p Black 456 4644 a([11])p Black 40 w(Y.)f(Puri.)p 0 1 0 0 TeXcolorcmyk 23 w Fc(A)n(rithmetic)h(Pr)l(op)l(erties)i(of)f (Perio)l(dic)i(Orbits)p Black [[185 160 359 172] [1 1 1 [3 3]] [0 0 1]] (http://www.mth.uea.ac.uk/admissions/graduate/theses/yash_puri/outline.pdf) pdfm 26 w Fn(.)24 b(PhD)g(thesis,)g(The)g(Univ)n(ersit)n(y)585 4744 y(of)j(East)g(Anglia,)g(2000.)p Black 456 4844 a([12])p Black 40 w(N.J.A.)h(Sloane.)p 0 1 0 0 TeXcolorcmyk 27 w Fc(Online)i(Encyclop)l(e)l(dia)i(of)f(Inte)l(ger)e(Se)l(quenc)l(es)p Black [[213 136 393 148] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/~njas/sequences) pdfm 29 w Fn(.)p Black 456 4943 a([13])p Black 40 w(Thomas)35 b(W)-7 b(ard.)35 b(Almost)h(all)g Fa(S)5 b Fn(-in)n(teger)34 b(dynamical)h(systems)g(ha)n(v)n(e)f(man)n(y)i(p)r(erio)r(dic)585 5043 y(p)r(oin)n(ts.)p 0 1 0 0 TeXcolorcmyk 27 w Fc(Er)l(go)l(dic)c (The)l(ory)f(Dynamic)l(al)g(Systems)p Black [[175 112 329 124] [1 1 1 [3 3]] [0 0 1]] (http://uk.cambridge.org/journals/ets/) pdfm 29 w Fn(,)d Fb(18)p Fn(:471{486)23 b(\(1998\).)p Black Black eop %%Trailer end end