Factorials and powers, a minimality result

TL;DR

This paper investigates the minimal positive integer n such that a^n < n! for a > 1, providing a method to determine this n and exploring related numerical sequences using advanced factorial approximations.

Contribution

It introduces a precise method to identify the smallest n satisfying a^n < n! for any a > 1, improving on existing factorial approximation techniques.

Findings

Derived a formula to compute the minimal n for given a > 1

Identified three key numerical sequences in the analysis

Enhanced factorial approximation using Robbins' refinement

Abstract

Let $a > 1$. Then $a^n < n!$ for some positive integer $n$. We show that the smallest such $n$ is one of a pair of possibilities, or is one possibility, which we show how to calculate. There are three interesting numerical sequences which play a central role in our arguments. This paper is based on the improvement on Sterling's approximation of factorials due to Robbins \cite{Robbins} and results of \cite{Radford}