JIPAM

Equations and Inequalities Involving $v_p(n!)$  
 
  Authors: Mehdi Hassani,  
  Keywords: Factorial function, Prime number, Inequality.  
  Date Received: 14/06/04  
  Date Accepted: 16/02/05  
  Subject Codes:

05A10, 11A41, 26D15, 26D20.

  
  Editors: László Tóth,  
 
  Abstract:

In this paper we study $ v_p(n!)$, the greatest power of prime $ p$ in factorization of $ n!$. We find some lower and upper bounds for $ v_p(n!)$, and we show that $ v_p(n!)=\frac{n}{p-1}+O(\ln n)$. By using the afore mentioned bounds, we study the equation $ v_p(n!)=v$ for a fixed positive integer $ v$. Also, we study the triangle inequality about $ v_p(n!)$, and show that the inequality $ p^{v_p(n!)}>q^{v_q(n!)}$ holds for primes $ p<q$ and sufficiently large values of $ n$. ;


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