On the prime power factorization of n!

In this paper we prove two results. The first theorem uses a paper of Kim [8] to show that for fixed primes p1, . . . , pk, and for fixed integers m1, . . . ,mk, with pi 6 |mi, the numbers (ep1(n), . . . , epk(n)) are uniformly distributed modulo (m1, . . . ,mk), where ep(n) is the order of the prime p in the factorization of n!. That implies one of Sander’s conjecture from [9], for any set of odd primes. Berend [1] asks to find the fastest growing function f(x) so that for large x and any given finite sequence εi ∈ {0, 1}, i ≤ f(x), there exists n < x such that the congruences epi(n) ≡ εi (mod 2) hold for all i ≤ f(x). Here, pi is the ith prime number. In our second result, we are able to show that f(x) can be taken to be at least c1(log x/(log log x)), with some absolute constant c1, provided that only the first odd prime numbers are involved.