Prime Pythagorean triangles

Harvey Dubner 449 Beverly Road, Ridgewood, New Jersey 07450

Tony Forbes

Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom

Email addresses: hdubner1@compuserve.com and tonyforbes@ltkz.demon.co.uk

Abstract

A prime Pythagorean triangle has three integer sides of which the hypotenuse and one leg are primes. In this article we investigate their properties and distribution. We are also interested in finding chains of such triangles, where the hypotenuse of one triangle is the leg of the next in the sequence. We exhibit a chain of seven prime Pythagorean triangles and we include a brief discussion of primality proofs for the larger elements (up to 2310 digits) of the associated set of eight primes.

1991 Mathematics Subject Classification: Primary 11A41 Keywords: Pythagorean triangles, prime numbers, primality proving

1. Introduction

While investigating the distribution of special forms of primes, the first author accidently came across a conjecture about Pythagorean triangles (right triangles with integral sides). The conjecture, based on the famous Conjecture (H) of Sierpiński and Schinzel, states that there is an infinite number of Pythagorean triangles which have a leg and hypotenuse both prime [9, page 408].

Pythagorean triangles have been the subject of much recreational material [1] as well as the basis of some of the most important and fundamental topics in number theory. However, we could not find any significant references to such two-prime Pythagorean triangles, and hoping that we had found a new topic to study we enthusiastically started (1) developing appropriate theory and computer programs;

(2) searching for large two-prime triangles;

(3) searching for sequences of two-prime triangles where the hypotenuse of the previous triangle becomes the leg of the next one.

The largest two-prime Pythagorean triangle that was found had a leg of 5357 digits and an hypotenuse of 10713 digits. It soon became apparent that finding sequences of triangles was exceptionally interesting and challenging. Eventually a sequence of seven triangles was found. More significant than the seven triangles is the improvement by the second author of the general method, APRCL, for primality proving so that the seventh hypotenuse of 2310 digits could be proved prime.

2. Theory

A two-prime Pythagorean triangle, A^{2} + B^{2} = C^{2}, must be primitive, so that

A = u^{2} − v^{2},  B = 2uv,  C = u^{2} + v^{2},

with gcd(u, v) = 1, and u, v of different parity. Since A = (u + v)(u − v), for A to be prime it is necessary that (u − v) = 1 so that

A = 2v + 1,  B = 2v^{2} + 2v,  C = 2v^{2} + 2v + 1.

Thus

(2.1)  C = \frac{A^{2} + 1}{2}.

Note that the even leg is only one less than the hypotenuse. The triangles get quite thin as A increases.

To find two-prime Pythagorean triangles it is necessary to find pairs of primes A, C that satisfy the above equation. Table 1 lists the smallest two-prime Pythagorian triangles.

Table 1. Pythagorean triangles with two prime sides

rank prime leg  even leg hypotenuse
1  3  4  5
2  5  12  13
3  11  60  61
4  19  180  181
5  29  420  421
6  59  1740  1741
7  61  1860  1861
8  71  2520  2521
9  79  3120  3121
10  101  5100  5101
100  4289  9197760  9197761
1000  91621 4197203820 4197203821

Small triangles are easy to find by a simple search, but finding large triangles with thousands of digits is complicated by the difficulty of proving true primality of the hypotenuse, C. However, if (C − 1) has many factors then it is easy to prove primality using [2], assuming that the factored part of (C − 1) exceeds \sqrt[3]{C}. Since

(2.2)  C − 1 = \frac{A^{2} + 1}{2} − 1 = (A^{2} − 1)/2 = (A − 1)(A + 1)/2,

by picking an appropriate form for A, then (A − 1) can be completely factored so that (C − 1) will be about 50% factored.

Using the form A = k·10^{n}+1, a computer search of a few days gave the following large triangle:

A = 491140 · 10^{1300} + 1, 1306 digits, C = 2612 digits.

A few days after this result was posted to the NMBRTHRY list we received a message from Iago Camboa announcing a much larger triangle:

A = 1491 · 2^{17783} + 1, 5357 digits, C = 10713 digits.

He cleverly used a previously computed list of primes as a source for A thus eliminating the large amount of time required to find the first prime.

3. Two-prime Pythagorean triangle sequences

It is possible to find a series of primes, P_{0}, P_{1}, P_{2}, ..., P_{k}, ..., P_{n} such that

(3.1)  P_{k+1} = \frac{P_{k}^{2} + 1}{2}.

This represents a sequence of n two-prime triangles where P_{k} is the hypotenuse of the k-th triangle and the leg of the (k + 1)-th triangle. Each P has about twice the number of digits as the previous P. Table 2 is a list of the smallest sets of two sequential prime Pythagorean triangles.

Table 2. Two sequential prime Pythagorean triangles

triangle 1  triangle 2
1  3  4  5  5  12  13
2 11  60  61  61  1860  1861
3 19  180  181  181  16380  16381
4 59  1740  1741  1741  1515540  1515541
5 271 36720 36721 36721  674215920  674215921
6 349 60900 60901 60901 1854465900 1854465901
7 521 135720 135721 135721 9210094920 9210094921
8 929 431520 431521 431521 93105186720 93105186721

Table 3 is a list of the starting primes for the smallest prime Pythagorean sequences for two, three, four and five triangles. These were found by straight forward unsophisticated searching and took about 10 computer-days (Pentium/200), mostly for finding five triangles.

Finding the starting prime for the smallest prime sequence of six triangles took about 120 computer days.

P_{0} for 6 triangles = 2500282512131.

Table 3. Starting prime for smallest prime Pythagorean sequences

2 triangles 3 triangles 4 triangles 5 triangles
1  3  271  169219  356498179
2  11  349  1370269  432448789
3  19  3001  5965699 5380300469
4  59  10099 15227879 10667785241
5  271  11719 17750981 11238777509
6  349  12281 19342559 12129977791
7  521  25889 21828601 23439934621
8  929  39901 24861761 28055887949
9  1031  46399 27379621 33990398249
10  1051  63659 34602049 34250028521
11  1171  169219 39844619 34418992099
12  2381  250361 48719711 34773959159
13  2671  264169 50049281 34821663421
14  2711  287629 51649019 36624331189
15  2719  289049 52187371 40410959231
16  3001  312581 52816609 43538725229
17  3499  353081 58026659 47426774869
18  3691  440681 73659239 48700811941
19  4349  473009 79782821 49177751131
20  4691  502501 86569771 59564407571

Next, we attempted to derive the number of n triangle sequences that could be expected. If the (n + 1) numbers that make up the n triangles were selected randomly but were of the proper size then the probability that P is the start of n triangles is

(3.2)  Q(P, n) = ∏_{0}^{n} \frac{1}{log P_{i}} = ∏_{0}^{n} \frac{1}{2^{i}(log P)} = \frac{1}{2^{n(n+1)/2}(log P)^{n+1}} .

However, there are correlations between the primes that affect the prime probabilities. It is easy to show from equation (2.1) that P_{0} can only end in 1 or 9, which elininates half the possible P_{0}’s, and assures that all subsequent potential primes cannot be divisible by 2, 3 or 5. Thus, the probability of each subsequent number being prime is increased by the factor (2/1)(3/2)(5/4) = 3.75. The probability that P is the start of n prime triangles now becomes,

(3.3)  Q(P, n) = \frac{0.5(3.75)^{n}}{2^{n(n+1)/2}(log P)^{n+1}} .

The expected number of prime triangles up to a given P_{0} is

(3.4)  E(P_{0}, n) = ∑_{P =3}^{P_{0}} Q(P, N) = \frac{0.5(3.75)^{n}}{2^{n(n+1)/2}} ∑_{P =3}^{P_{0}} \frac{1}{(log P)^{n+1}} .

The last summation can be approximated by an integral, which after integrating by parts becomes,

R(P, n) = \frac{1}{n!}Li(P) − \frac{1}{n!} \frac{P}{log P} − ⋯ − \frac{1}{n(n − 1)} \frac{P}{(log P)^{(n−1)}} − \frac{1}{n} \frac{P}{(log P)^{n}} ,

where Li(P) is the logarithmic integral. Equation (3.4) now becomes

(3.5)  E(P_{0}, n) = \frac{0.5(3.75)^{n}}{2^{n(n+1)/2}} R(P_{0}, n)(1.3)^{n} .

Note the inclusion of a correction factor, (1.3)^{n}. As is discussed in the following section on sieving, there are other correlations between the primes which affect the expectation. These are difficult to derive theoretically so we determined it empirically. Table 4 compares the estimated and actual number of triangles found. The corrected estimate appears adequate to assist in estimating the search time for seven prime Pythagorean triangles.

Table 4. Estimated and actual number of prime Pythagorean triangles

triangles  corrected
n  P_{0}  actual estimate estimate
1  130000 1302  1090  1420
2  1980000 1005  741  1252
3  10^{8}  953  469  1030
4  18 · 10^{8}  205  53  151
5  63 · 10^{9}  21  4  15
6  28 · 10^{12}  1  0.14  0.7

Next, we use equation (3.5) to estimate the smallest P_{0} that will give seven triangles. The following table shows we can expect that P_{0} for seven triangles will be about 6700 times larger than P_{0} for six triangles. Using performance data from the search for six triangles, this means that the search for the smallest sequence of seven prime Pythagorean triangles could be expected to take about 200 computeryears!

n P_{0} for expectation=1 actual P_{0}
2 28  3
3 1,350  271
4 1,000,000  169, 219
5 1.5 · 10^{9}  3.5 · 10^{8}
6 4.0 · 10^{12}  2.5 · 10^{12}
7 2.7 · 10^{16}

It was clear that the search for the smallest sequence of seven triangles as presently constituted was impractical. For every P_{0} the search method included testing by division to see if each of the (n + 1) potential primes was free of small factors. The second author then proposed an efficient sieving method that limited the search to sequences that had a high probability of success. This made a search for seven triangles reasonable.

4. The Sieve

A set of seven Pythagorean triangles with the desired properties is equivalent to a chain of eight primes, P_{0}, P_{1}, …, P_{7}, linked by the condition P_{i+1} = (P_{i}^{2} + 1)/2, i = 0, 1, …, 6.

The purpose of the sieve is to eliminate from further consideration numbers P_{0} for which either P_{0} itself or one of the numbers P_{i}, i = 1, 2, …, 7, is divisible by a small prime. Let q be an odd prime and suppose P is to be considered as a possible value of P_{0}. Clearly, we can reject P if P ≡ 0 (mod q). Furthermore, we can reject P if P_{1} is divisible by q, that is, if

P ≡ \sqrt{−1} (mod q),

on the assumption that (\frac{−1}{q}) = 1. Continuing in this way, we can reject P if

P ≡ \sqrt{2\sqrt{−1} − 1} (mod q)

(for then P_{2} is divisible by q), or if

P ≡ \sqrt{2\sqrt{2\sqrt{−1} − 1} − 1} (mod q),

and so on, provided that the various square roots (mod q) exist. In each case, where there is a square root (mod q) there are two possible values and hence two extra residues (mod q) that can be eliminated.

For prime q, we compute the set E(q) of forbidden residues (mod q) as follows. Start with E_{0}(q) = {0}. Given E_{i}(q), let

E_{i+1} = \{±\sqrt{2e − 1} (mod q) : e ∈ E_{i} and (\frac{2e − 1}{q}) = 1\} .

Then E(q) is the union of E_{0}(q), E_{1}(q), …, E_{7}(q). In Table 5 we list E(q) for the first few primes q ≡ 1 (mod 4).

Now let

P = NQ + H,

where Q is the product of small primes and H is allowed to run through all the permitted residues (mod Q). We sieve the numbers N. That is, we start with an interval N_{0} ≤ N < N_{1} and for each sieving prime q, gcd(q, Q) = 1, we remove all those N ∈ [N_{0}, N_{1}) for which NQ + H is divisible by q.

We split Q into pairwise coprime divisors m_{0}, m_{1}, …, m_{r}. For each divisor m_{j} of Q, j = 0, 1, …, r, we make a list of the permitted residues (mod m_{j}); h is a permitted residue (mod m_{j}) if h is not zero (mod m_{j}) and if the function h → (h^{2} + 1)/2 (mod m_{j}) does not produce zero (mod m_{j}) during the first seven iterations. The permitted residues H (mod Q) are constructed from permitted residues h (mod m_{j}) using the Chinese Remainder Theorem. It works well if Q is the product of primes which have small percentages of permitted residues. From this perspective the best primes, in descending order of merit, turn out to be: 29 (34%), 5 (40%), 2 (50%), 17 (59%), 13 (62%), 3 (67%), 53 (68%), 101 (71%), 89 (74%) and 233 (77%).

For the actual search we chose Q = 21342962305470, with divisors 6630 = 2 · 3 · 5 · 13 · 17, 29, 89, 101, 53, and 233. The number of values of H (mod Q) turns out to be 320 · 10 · 66 · 72 · 36 · 180 = 98537472000, the indicated factors of this product being the numbers of permitted residues modulo the corresponding factors of Q.

The construction of the sieve and the method of computing H (mod Q) were based on computer programs designed for finding prime k-tuplets; see [6] for the details. We set up a table of sieving primes q together with pre-computed values of −1/Q (mod q) as well as, for q ≡ 1 (mod 4), e/Q (mod q) for each pair e, q − e in E(q). We can then rapidly calculate the index of the first N to be removed from the sieve array for P ≡ e (mod q): e/Q − H/Q − N_{0} (mod q).

The program also allows us to limit the size of primes q ≡ 3 (mod 4) used by the sieve. One reason for doing so would be to give priority to primes q ≡ 1 (mod 4); they have more residues for sieving and therefore one would expect them to be in some sense more efficient. In fact it was found by experiment that if P has about 19 digits, a sieve limit L_{0} = 20000 for q ≡ 3 (mod 4) and 480000 for q ≡ 1 (mod 4) was approximately optimal.

Further performance improvements are possible by limiting the influence of primes q ≡ 1 (mod 4). For each P that survives the sieve we do a probableprimality test, 2^{P} ≡ 2 (mod P), on P as well as, if P turns out to be a probableprime, the numbers that follow P in the chain, stopping as soon as a composite is found. The effort required to perform the probable-primality test increases by a factor of about eight as we move from P_{i} to P_{i+1}. Therefore it might be better if priority were given to sieving primes q and residues e (mod q) which would eliminate composite numbers from the larger elements of the chain.

For controlling the effect of primes q ≡ 1 (mod 4) we provided a set of parameters L_{1}, L_{2}, . . .. If q ≡ 1 (mod 4) is a sieving prime and e ∈ E_{i}(q) then we do not use residue e (mod q) for sieving unless q < L_{i}. As a result of a certain amount of experimentation we found that the optimum sieving rate occurs with the limits set approximately as follows: L_{1} = 120000, L_{2} = 240000, L_{3} = 360000, together with a limit L_{0} = 20000 for primes ≡ 3 (mod 4) and an overall sieve limit of 480000. From these values we can compute an expected survival rate of

∏_{q prime} \frac{q − ν_{q}}{q} = \frac{1}{3770},

approximately, where ν_{q} is the number of residues (mod q) used by the sieve.

5. Eight Primes

In September 1999 the search was successful and this chain of eight probable primes was found:

P_{0} = 2185103796349763249  (19 digits),
P_{1} = (P_{0}^{2} + 1)/2  (37 digits),
P_{2} = (P_{1}^{2} + 1)/2  (73 digits),
P_{3} = (P_{2}^{2} + 1)/2  (145 digits),
P_{4} = (P_{3}^{2} + 1)/2  (289 digits),
P_{5} = (P_{4}^{2} + 1)/2  (579 digits),
P_{6} = (P_{5}^{2} + 1)/2  (1155 digits),
P_{7} = (P_{6}^{2} + 1)/2  (2310 digits).

The search program was designed to run on standard IBM PCs. We employed about 15 such machines, with clock speeds ranging from 200 MHz to 400 MHz. The faster computers were sieving and testing numbers at rates of about ten billion per hour. We also found 174 additional chains of seven (probable) primes.

6. Primality Proofs

The first six numbers, P_{0}, P_{1}, …, P_{5}, as well as other small primes mentioned in this section are easily verified by the UBASIC [3] program APRT-CLE, a straightforward implementation of the APRCL test. For i = 6 and 7 we attempt to factorize

P_{i} − 1 = (P_{0} − 1) ∏_{j=0}^{i−1} \frac{1}{2} (P_{j} + 1) .

Thus

P_{0} − 1 = 2^{4} · 233 · 586132992583091,
(P_{0} + 1)/2 = 3^{2} · 5^{3} · 13 · 761 · 19087 · 5143087,
(P_{1} + 1)/2 = 7^{2} · 1063 · 189043 · 7552723 · 113558719 · 141341652553,
(P_{2} + 1)/2 = 7058053 · 5848063479673576700713235221
·34520041584369005634844907730019249777,
(P_{3} + 1)/2 = 2179 · 1847645923 · C_{132},
(P_{4} + 1)/2 = 307 · 769 · 262513 · P_{278},
(P_{5} + 1)/2 = 108139 · 11360649709 · 5586562264501 · C_{550},
(P_{6} + 1)/2 = 4177 · 1372052449 · 5098721569 · 84098816095916212867 · C_{1113},

where C_{132}, C_{550} and C_{1113} are composite numbers of 132, 550 and 1113 digits, respectively, and P_{278} is a 278-digit prime:

P_{278} = 66505518540598996114987486506055236521044267373138

69473288000457727001877127498646545001634613677898 53932112480508999228232340454335875401889420451888 17780482079524485531037464472393979852934170207932 02663155485302406204947222346461607409301255277393 4788467292248055697961196019.

The 28-digit factor of P_{2} +1 and the 20-digit factor of P_{6} +1 were found by Manfred Toplic and Paul Zimmermann.

Since we have a 41% partial factorization of P_{6} −1 we can establish the primality of P_{6} by the methods of Brillhart, Lehmer and Selfridge [2]. (Similarly a 77% factorization of P_{5} − 1 provides an alternative proof for P_{5}.)

It remains to deal with P_{7}. We do not have enough prime factors of P_{7} − 1 for a simple proof, so we use a combination of methods. Suppose d < P_{7} is a prime factor of P_{7}. The proof that no such d exists proceeds in several stages.

Gathering together the prime factors of P_{7} listed above, let

F_{1} = 11364028773118678645863393880225035110068188490680

74284625807644534721210969640169863192044176288720 57382836214336492569310719940321645143241641366672 31704620613678520580684280352992373327229897947340 09917692032743575475918022578947700337216860293874 96561498464943981086970289943873321681460108830000 00131801406514260770804840415255291401064877989705 76202962420323563098312300324091122817224414751412 15123765209184430598589590008879997663256918503367 07250451432160496252649191808276871593840887080642 91103468534974000 (517 digits).

Then, after confirming that the conditions of Pocklington’s theorem [8] hold, we have

(6.1)  d ≡ 1 (mod F_{1}).

Similarly, by Morrison’s theorem [2, Theorem 16],

(6.2)  d ≡ ±1 (mod F_{2}),

where F_{2} = 43^{2} · 73 = 134977.

Next, we confirm that the conditions for the APRCL test (see, for example, Cohen and A. K. Lenstra [4] or Cohen and H. W. Lenstra [5]) are satisfied with the prime powers p^{k}: {2^{5}, 3^{3}, 5^{2}, 7, 11, 13}, and primes q: {11, 17, 19, 23, 29, 31, 37, 41, 53, 61, 67, 71, 79, 89, 97, 101, 109, 113, 127, 131, 151, 157, 181, 199, 211, 241, 271, 281, 313, 331, 337, 353, 379, 397, 401, 421, 433, 463, 521, 541, 547, 601, 617, 631, 661, 673, 701, 757, 859, 881, 911, 937, 991, 1009, 1051, 1093, 1171, 1201, 1249, 1301, 1321, 1801, 1873, 1951, 2003, 2017, 2081, 2161, 2311, 2341, 2377, 2521, 2731, 2801, 2861, 2971, 3121, 3169, 3301, 3361, 3433, 3511, 3697, 3851, 4159, 4201, 4621, 4951, 5281, 5851, 6007, 6301, 6553, 7151, 7393, 7561, 7723, 8009, 8191, 8317, 8581, 8737, 9241, 9829, 9901, 11551, 11701, 12601, 13729, 14561, 14851, 15121, 15401, 15601, 16381, 16633, 17551, 18481, 19801, 20021, 20593, 21601, 21841, 23761, 24571, 25741, 26209, 28081, 30241, 34651, 36037, 38611, 39313, 42901, 47521, 48049, 50051, 51481, 54601, 55441, 65521, 66529, 70201, 72073, 79201, 81901, 92401, 93601, 96097, 103951, 108109, 109201, 110881, 118801, 120121, 123553, 131041, 140401, 150151, 151201, 180181, 193051, 196561, 200201, 216217, 218401, 257401, 270271, 300301, 332641, 393121, 415801, 432433, 450451}. The result is that

(6.3)  d ≡ P_{7}^{i} (mod S) for some i = 1, 2, …, T − 1,

where T = 21621600 is the product of the p^{k}s and S = 8.164364 · 10^{634}, approximately, is the product of the qs.

Let G = F_{1}F_{2}S and observe that F_{1}, F_{2} and S are pairwise coprime. We combine (6.1), (6.2) and (6.3) by the Chinese Remainder Theorem to obtain

d ≡ (\frac{1}{F_{2}S} mod F_{1}) F_{2}S + (\frac{e}{F_{1}S} mod F_{2}) F_{1}S

+ (\frac{P_{7}^{i}}{F_{1}F_{2}} mod S) F_{1}F_{2} (mod G)

for some e = ±1 and i = 1, 2, …, T −1. After eliminating all possible d < \sqrt{P_{7}} < G by trial division we can conclude that P_{7} is prime.

7. Acknowledgements

We would like to thank Jeremy Humphries, Manfred Toplic and Paul Zimmermann for contributing their own computer resources to the search for seven prime Pythagorean triangles. We are specifically grateful to Paul Zimmermann, who also made his Elliptic Curve program available to us for the partial factorization of P_{7} − 1.

References

1. A. H. Beiler, Recreations In the Theory of Numbers, 2nd ed., Dover Publications, New York, ch. XIV, 1966.

2. John Brillhart, D. H. Lehmer and J. L. Selfridge, New primality criteria and factorizations of 2^{m} ± 1, Math. Comp., 29 (1975), 620-647.

3. C. K. Caldwell, UBASIC, J. Recreational Math., 25 (1993), 47-54.

4. H. Cohen and A. K. Lenstra, Implementation of a new primality test, Math. Comp., 48 (1987), 103-121.

5. H. Cohen and H. W. Lenstra, Primality testing and Jacobi sums, Math. Comp., 42 (1984), 297-330.

6. Tony Forbes, Prime clusters and Cunningham chains, Math. Comp., 68 (1999), 1739-1747.

7. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford University Press, 1979.

8. H. C. Pocklington, The determination of the prime or composite nature of large numbers by Fermat’s theorem, Proc. Cambridge Philos. Soc., 18 (1914-16), 29-30.

9. P. Ribenboim, The New Book of Prime Number Records, 3rd ed., Springer-Verlag, New York, 1995.

(Mentions sequences A048161, A048270 and A048295.)

Received May 6, 2001; revised version received Sept. 3, 2001. Published in Journal of Integer Sequences Sept. 13, 2001.

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Table 5. E(q) ; \begin{aligned}q & E(q) \\ 5 & \{0, 2, 3\} \\ 13 & \{0, 3, 5, 8, 10\} \\ 17 & \{0, 3, 4, 5, 12, 13, 14\} \\ 29 & \{0, 2, 3, 5, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 24, 26, 27\} \\ 37 & \{0, 6, 8, 14, 23, 29, 31\} \\ 41 & \{0, 9, 32\} \\ 53 & \{0, 4, 14, 16, 17, 18, 19, 22, 23, 30, 31, 34, 35, 36, 37, 39, 49\} \\ 61 & \{0, 11, 50\} \\ 73 & \{0, 23, 27, 46, 50\} \\ 89 & \{0, 9, 26, 27, 30, 34, 37, 38, 39, 40, 41, 44, 45, 48, 49, 50, 51, 52, \\ & 55, 59, 62, 63, 80\} \\ 97 & \{0, 7, 22, 25, 72, 75, 90\} \\ 101 & \{0, 7, 10, 12, 15, 16, 22, 23, 25, 26, 34, 35, 37, 38, 50, 51, 63, 64, \\ & 66, 67, 75, 76, 78, 79, 85, 86, 89, 91, 94\} \\ 109 & \{0, 33, 76\} \\ 113 & \{0, 2, 15, 46, 54, 59, 67, 98, 111\} \\ 137 & \{0, 22, 37, 100, 115\} \\ 149 & \{0, 44, 105\} \\ 157 & \{0, 10, 28, 31, 126, 129, 147\} \\ 173 & \{0, 32, 80, 93, 141\} \\ 181 & \{0, 2, 9, 19, 30, 33, 41, 47, 54, 56, 64, 78, 80, 88, 93, 101, 103, 117, \\ & 125, 127, 134, 140, 148, 151, 162, 172, 179\} \\ 193 & \{0, 57, 81, 112, 136\} \\ 197 & \{0, 14, 37, 94, 103, 160, 183\} \\ 229 & \{0, 18, 19, 30, 48, 54, 59, 69, 91, 107, 110, 119, 122, 138, 160, 170, \\ & 175, 181, 199, 210, 211\} \\ 233 & \{0, 3, 5, 7, 12, 13, 16, 21, 25, 27, 30, 42, 43, 44, 48, 52, 53, 55, 61, \\ & 67, 71, 80, 85, 89, 101, 104, 115, 118, 129, 132, 144, 148, 153, 162, 166, \\ & 172, 178, 180, 181, 185, 189, 190, 191, 203, 206, 208, 212, 217, 220, 221, \\ & 226, 228, 230\} \\ 241 & \{0, 64, 177\} \\ 257 & \{0, 16, 51, 206, 241\} \\ 269 & \{0, 82, 187\} \\ 277 & \{0, 8, 52, 60, 106, 171, 217, 225, 269\} \\ 281 & \{0, 53, 228\} \\ 293 & \{0, 4, 121, 138, 155, 172, 289\} \\ 313 & \{0, 7, 21, 25, 92, 221, 288, 292, 306\} \\ 317 & \{0, 17, 23, 24, 31, 44, 50, 52, 56, 74, 97, 114, 115, 126, 130, 134, \\ & 141, 142, 145, 153, 164, 172, 175, 176, 183, 187, 191, 202, 203, 220, 243, \\ & 261, 265, 267, 273, 286, 293, 294, 300\} \\ 337 & \{0, 21, 31, 34, 50, 71, 73, 90, 110, 114, 116, 144, 148, 153, 157, 162, \\ & 175, 180, 184, 189, 193, 221, 223, 227, 247, 264, 266, 287, 303, 306, 316\} \end{aligned}