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  Volume 6, Issue 2, Article 29
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,  

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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