On the distance between factorials and repunits

TL;DR

This paper investigates the relationship between factorials and repunits, providing bounds and conditions under which their differences are positive or negative, contributing to understanding their numerical proximity.

Contribution

It establishes new lower bounds for the difference between factorials and repunits for large integers and primes, extending previous results in number theory.

Findings

Derived lower bounds for factorial and repunit differences

Identified conditions for positivity and negativity of the difference

Extended understanding of the numerical relationship between factorials and repunits

Abstract

We show that if $n\ge n_0$, $b\ge 2$ are integers, $p\ge 7$ is prime and $n!-(b^p-1)/(b-1)\ge 0$, then $n!-(b^p-1)/(b-1) \ge 0.5\log\log n/\log\log\log n$. Further results are obtained, in particular for the case $n!-(b^p-1)/(b-1) < 0$.