On the prime power factorization of n!, II Original Research Article

Let p, q be primes and m be a positive integer. For a positive integer n, let ep(n) be the nonnegative integer with pep(n)n and pep(n)+1does not dividen. The following results are proved: (1) For any positive integer m, any prime p and any εset membership, variantZm, there are infinitely many positive integers n such that image; (2) For any positive integer m, there exists a constant D(m) such that if ε,δset membership, variantZm and p, q are two distinct primes with max{p,q}greater-or-equal, slantedD(m), then there exist infinitely many positive integers n such that image, image. Finally we pose four open problems.