A New Generalization of Fermat's Last Theorem

2013/10/03 by Tianxin Cai, Cai, Tianxin, Deyi Chen +3
Computer Science · Mathematics · #11D41 #11D72 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1310.0897

openalex publication_date 2013/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider some hybrid Diophantine equations of addition and multiplication. We first improve a result on new Hilbert-Waring problem. Then we consider the equation \begincases A+B=C ABC=Dⁿ \endcases where A,B,C,D,n ∈\ZZ₊ and n≥3, which may be regarded as a generalization of Fermat's equation xⁿ+yⁿ=zⁿ. When gcd(A,B,C)=1, (1) is equivalent to Fermat's equation, which means it has no positive integer solutions. We discuss several cases for gcd(A,B,C)=p^k where p is an odd prime. In particular, for k=1 we prove that (1) has no nonzero integer solutions when n=3 and we conjecture that it is also true for any prime n>3. Finally, we consider equation (1) in quadratic fields ℚ(√(t)) for n=3.

Citations

• Modular Elliptic Curves and Fermat's Last Theorem

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