On the sizes of the maximal prime powers divisors of factorials

TL;DR

The paper proves that for large enough n, the maximal prime power dividing n! is greater for smaller primes than for larger primes, with explicit bounds for twin primes.

Contribution

It establishes a prime-dependent threshold n_0 beyond which the maximal prime power divisors of n! follow a specific order, including explicit results for twin primes.

Findings

Existence of a prime-dependent n_0 such that q^{ν_q(n!)} < p^{ν_p(n!)} for all n ≥ n_0 and q > p.

For twin primes p and q = p + 2, the minimal n_0 is explicitly given by (p^2 + p)/2.

The result applies uniformly across all primes q > p for sufficiently large n.

Abstract

Let p be any prime, and $p^(\nu_p(n!))$ the maximal power of $p$ dividing $n!$. It is proved that there exists a positive integer $n_0$, which depends only on $p$, such that $q^(\nu_q(n!)) < p^(\nu_p(n!))$ for all $n \ge n_0$ and all primes $q > p$. For twin primes $p$ and $q = p + 2$ it is proved that the minimal $n_0$ satisfying $q^(\nu_q(n!)) < p^(\nu_p(n!))$ for all $n \ge n_0$ is given by $n_0 = (p^2+p)/2$.