The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations
Number Theory
2017-07-25 v2 Formal Languages and Automata Theory
Combinatorics
Abstract
We consider a natural generalization of the Nagell-Ljunggren equation to the case where the qth power of an integer y, for q >= 2, has a base-b representation that consists of a length-l block of digits repeated n times, where n >= 2. Assuming the abc conjecture of Masser and Oesterl\'e, we completely characterize those triples (q, n, l) for which there are infinitely many solutions b. In all cases predicted by the abc conjecture, we are able (without any assumptions) to prove there are indeed infinitely many solutions.
Cite
@article{arxiv.1707.03894,
title = {The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations},
author = {Andrew Bridy and Robert J. Lemke Oliver and Arlo Shallit and Jeffrey Shallit},
journal= {arXiv preprint arXiv:1707.03894},
year = {2017}
}