On the prime power factorization of <f>n!</f>

In this paper, we prove two results. The first theorem uses a paper of Kim (J. Number Theory 74 (1999) 307) to show that for fixed primes <f>p1,…,pk</f>, and for fixed integers <f>m1,…,mk</f>, with <f>pi∤mi</f>, the numbers <f>(ep1(n),…,epk(n))</f> are uniformly distributed modulo <f>(m1,…,mk)</f>, where <f>ep(n)</f> is the order of the prime <f>p</f> in the factorization of <f>n!</f>. That implies one of Sander''s conjectures from Sander (J. Number Theory 90 (2001) 316) for any set of odd primes. Berend (J. Number Theory 64 (1997) 13) asks to find the fastest growing function <f>f(x)</f> so that for large <f>x</f> and any given finite sequence <f>&epsiv;i∈{0,1}, i&les;f(x)</f>, there exists <f>n<x</f> such that the congruences <f>epi(n)≡&epsiv;i (mod 2)</f> hold for all <f>i&les;f(x)</f>. Here, <f>pi</f> is the <f>i</f>th prime number. In our second result, we are able to show that <f>f(x)</f> can be taken to be at least <f>c1(log x/(log log x)6)1/9</f>, with some absolute constant <f>c1</f>, provided that only the first odd prime numbers are involved. [Copyright &y& Elsevier]