Equations and Inequalities Involving $v_p(n!)$

摘要

In this paper we study , the greatest power of prime in factorization of . We find some lower and upper bounds for , and we show that . By using the afore mentioned bounds, we study the equation for a fixed positive integer . Also, we study the triangle inequality about , and show that the inequality q^{v_q(n!)}$ --> holds for primes and sufficiently large values of .