Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 346.10004
Autor: Erdös, Paul; Graham, Ronald L.
Title: On products of factorials. (In English)
Source: Bull. Inst. Math., Acad. Sinica 4, 337-355 (1976).
Review: In this paper products of factorials are studied. Let a1 < a2 < ... < at = n be positive integers. First it is proved that the number of distinct integers of the form prod tk = 1ak! is \exp {(1+0(1))n log log n/ log n }. The main part of the paper is devoted to the Diophantine equation
prod tk = 1ak! = y2, (*) where t is fixed. Put F(t) = {n | (*) is solvable } and D(t) = F(t)-F(t-1). It is shown that D(1) = {1 }, D(2) = {n2: n > 1 }, D(3) \ne Ø, D(3) has density 0, D(4) has positive lower density, D(5) \ne Ø, 527 is the smallest element of D(6), D(t) = Ø for t > 6. Numerous other results are given. For example, n in F(6) if and only if n is composite. If n = m2r with m > 1, then n in F(4). If p \leq 11 is a proper prime divisor of n, then n in F(5). For almost all primes p one has 13 p \not in F(5). Some results from prime number theory are used in the proofs and quite a few open problems are mentioned.
Reviewer: R.Tijdeman
Classif.: * 11A41 Elemementary prime number theory
11D57 Multiplicative and norm form diophantine equations
11N05 Distribution of primes
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag