Counting solvable $\mathcal S$-unit equations and linear recurrence sequences with zeros

Alina Ostafe; Carl Pomerance; Igor E. Shparlinski

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arXiv · 2503.03985

Counting solvable $\mathcal S$-unit equations and linear recurrence sequences with zeros

Alina Ostafe·Carl Pomerance·Igor E. Shparlinski

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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.

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Alina Ostafe, Carl Pomerance, Igor E. Shparlinski. 2025-03-06. Counting solvable $\mathcal S$-unit equations and linear recurrence sequences with zeros. https://arxiv.org/abs/2503.03985

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