A refined modular approach to the Diophantine equation x²+y²ⁿ=z³

2010/01/29 by Sander R. Dahmen, Dahmen, Sander R. · 2 citations
Arts and Humanities · Mathematics · #11D41 #11F11 #11F80 #11G05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #French Literature and Poetry #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1002.0020

openalex publication_date 2010/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let n be a positive integer and consider the Diophantine equation of generalized Fermat type x²+y²ⁿ=z³ in nonzero coprime integer unknowns x,y,z. Using methods of modular forms and Galois representations for approaching Diophantine equations, we show that for n ∈ \5, 31\ there are no solutions to this equation. Combining this with previously known results, this allows a complete description of all solutions to the Diophantine equation above for n ≤ 10⁷. Finally, we show that there are also no solutions for n≡ -1 \pmod6.

Citations

• Modular Elliptic Curves and Fermat's Last Theorem

Cited by

• The inter-universal Teichmüller theory and new Diophantine results over rational numbers. II
• Construction of Bhaskara Pairs

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