Counting solvable $\mathcal S$-unit equations and linear recurrence sequences with zeros
Number Theory
2025-03-07 v1
Abstract
We show that only a rather small proportion of linear equations are solvable in elements of a fixed finitely generated subgroup of a multiplicative group of a number field. The argument is based on modular techniques combined with a classical idea of P. Erd\H{o}s (1935). We then use similar ideas to get a tight upper bound on the number of linear recurrence sequences which attain a zero value.
Cite
@article{arxiv.2503.03985,
title = {Counting solvable $\mathcal S$-unit equations and linear recurrence sequences with zeros},
author = {Alina Ostafe and Carl Pomerance and Igor E. Shparlinski},
journal= {arXiv preprint arXiv:2503.03985},
year = {2025}
}