A robust implementation for solving the S-unit equation and several applications

2019/03/03 by Alejandra Alvarado, Alvarado, Alejandra, Angelos Koutsianas +9 · 2 citations
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1903.00977

openalex publication_date 2019/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a number field, and S a finite set of places in K containing all infinite places. We present an implementation for solving the S-unit equation x + y = 1, x,y ∈\mathscrO_K,S^× in the computer algebra package SageMath. This paper outlines the mathematical basis for the implementation. We discuss and reference the results of extensive computations, including exponent bounds for solutions in many fields of small degree for small sets S. As an application, we prove an asymptotic version of Fermat's Last Theorem for totally real cubic number fields with bounded discriminant where 2 is totally ramified. In addition, we use the implementation to find all solutions to some cubic Ramanujan-Nagell equations.

Cited by

• Effective Generalized Fermat equation of signature (2p, 2q, r) with odd narrow class number
• On Fermat's Last Theorem over the ℤ₃-extension of ℚ and other fields

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