On the prime power factorization of n! |
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Authors: | Florian Luca |
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Institution: | a Instituto de Matematicas de lar UNAM, Campus Morelia, Apartado Postal 61-3 (Xangari) CP 58 089, Morelia, Michoacan, Mexico b Department of Mathematics, Auburn University Montgomery, PO Box 244023, Montgomery, AL 36124-4023, USA |
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Abstract: | 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 p1,…,pk, and for fixed integers m1,…,mk, with , 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 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(x) so that for large x and any given finite sequence , there exists n<x such that the congruences 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 , with some absolute constant c1, provided that only the first odd prime numbers are involved. |
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