| Let p1,p2,…be the sequence of all primes in ascending order. For a positive integer n, let epi(n) be the nonnegative integer with andIn 1980, Erdos and Graham posed the following conjecture: For every positive integer k there exist infinitely many positive integers n such that are all even.In 1997 D.Berend proved Erdos and Graham Conjecture. Several authors, Y. G. Chen and Y. C. Zhu, J. W. Sander, Y. G. Chen and F. Luca and P. Stǎnicǎ, continued working on this problem, each obtaining a stronger version of Erd(o|¨)s and Graham Conjecture.In this paper, we investigate some problems on the exponents modulo m in the standard factorization of n!. The main results are summarized as follows.1. Let p be a prime and m be a positive integer. It is showed that for any given prime p and any given positive integer m, ep(n!) is well distributed modulo m. In fact, Sander proved the result with modulus 2. The result with modulus 3 is pressed in Bull.Austral. Math.Soc.. Another result is that we get a better error term for the asymptotic formula for the counting function with regard to the exponents of prime p modulo p in the standard factorization of n!.2. Let p, q be primes and m be a positive integer. In this dissertation, the following result is pressed in J. Number Theory : For any positive integer m, there exists a constant D(m) such that ifε,δ∈(?)m and p, q are two distinct primes with max{p, q}≥D(m), then there exist infinitely many positive integers n such thatep(n!) =ε(mod m), eq(n!) =δ(mod m). |
