# A proof of Sondow's conjecture on the Smarandache function

## TL;DR

This paper proves Sondow's conjecture by showing that for any fixed k > 1, n^k < S(n)! holds for almost all positive integers n, confirming the inequality for the Smarandache function.

## Contribution

The paper provides a proof that the inequality n^2 < S(n)! holds for almost all positive integers, confirming Sondow's conjecture.

## Key findings

- The inequality n^k < S(n)! holds for almost all n for any fixed k > 1.
- Sondow's conjecture n^2 < S(n)! is confirmed for almost all positive integers.
- The result extends the understanding of the growth of the Smarandache function.

## Abstract

The Smarandache function of a positive integer $n$, denoted by $S(n)$, is defined to be the smallest positive integer $j$ such that $n$ divides the factorial $j!$. In this note, we prove that for any fixed number $k > 1$, the inequality $n^k < S(n)!$ holds for almost all positive integers $n$. This confirms Sondow's conjecture which asserts that the inequality $n^2 < S(n)!$ holds for almost all positive integers $n$.