The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations

TL;DR

This paper generalizes the Nagell-Ljunggren problem to powers with repetitive digit patterns in a given base, characterizing when infinitely many solutions exist under the abc conjecture and proving their existence unconditionally in certain cases.

Contribution

It extends the classical problem to a broader setting involving base-b digit repetitions and provides a complete characterization of solutions assuming the abc conjecture, with unconditional proofs in some cases.

Findings

Complete characterization of (q, n, l) triples with infinitely many solutions under the abc conjecture.

Unconditional proof of infinitely many solutions in all cases predicted by the conjecture.

Generalization of the Nagell-Ljunggren problem to powers with repetitive base-b representations.

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.