On a conjecture concerning the number of solutions to a^x+b^y=c^z

2022/06/28 by LE Mao-hua, Reese Scott, Le, Maohua +3
Mathematics · #11D61 #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2206.14032

openalex publication_date 2022/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let a, b, c be fixed coprime positive integers with min\ a,b,c \ &gt;1. Let N(a,b,c) denote the number of positive integer solutions (x,y,z) of the equation a^x + b^y = c^z. We show that if (a,b,c) is a triple of distinct primes for which N(a,b,c)&gt;1 and (a,b,c) is not one of the six known such triples then, taking a<b>10¹⁴. These results support a conjecture of the last two authors.</b>

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