On the distance between factorials and repunits
Volume 221 / 2025
Acta Arithmetica 221 (2025), 329-354
MSC: Primary 11D61
DOI: 10.4064/aa241115-29-7
Published online: 21 October 2025
Abstract
We show that if $n\ge n_0$ and $b\ge 2$ 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 $n!-(b^p-1)/(b-1) \lt 0$.
