On some conjectures of exponential Diophantine equations

2020/12/31 by Hairong Bai, Bai, Hairong
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2012.15401

openalex publication_date 2020/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the exponential Diophantine equation a^x+b^y=c^z, where a, b, c be relatively prime positive integers such that a²+b²=c^r, r∈ Z⁺, 2| r with b even. That is a=| Re(m+n√(-1))^r|, b=| Im(m+n√(-1))^r|, c=m²+n², where m, n are positive integers with m>n, m-n≡1(mod 2), gcd(m, n)=1. (x, y, z)= (2, 2, r) is called the trivial solution of the equation. In this paper we prove that the equation has no nontrivial solutions in positive integers x, y, z when r≡ 2(mod 4), m≡ 3(mod 4), mgt;max\n^10.4×10¹¹log(5.2×10¹¹log n), 3e^r, 70.2nr\. Especially the equation has no nontrivial solutions in positive integers x, y, z when r=2, m≡ 3(mod 4), mgt;n^10.4×10¹¹log(5.2×10¹¹log n).

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