On the prime power factorization of n!, II

Abstract

Let p, q be primes and m be a positive integer. For a positive integer n, let e p ( n ) be the nonnegative integer with p e p ( n ) | n and p e p ( n ) + 1 ∤ n . The following results are proved: (1) For any positive integer m, any prime p and any ε ∈ Z m , there are infinitely many positive integers n such that e p ( n ! ) ≡ ε ( mod m ) ; (2) For any positive integer m, there exists a constant D ( m ) such that if ε , δ ∈ Z m and p, q are two distinct primes with max { p , q } ⩾ D ( m ) , then there exist infinitely many positive integers n such that e p ( n ! ) ≡ ε ( mod m ) , e q ( n ! ) ≡ δ ( mod m ) . Finally we pose four open problems.