On the distance 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 $n!-(b^p-1)/(b-1)<0$.