An application of Baker's method to the Jeśmanowicz' conjecture on primitive Pythagorean triples

2018/11/01 by LE Mao-hua, Le, Maohua
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.1811.00654

openalex publication_date 2018/11/01 · openalex created_date 2018/11/09 · openalex updated_date 2026/07/28

Abstract

Let m, n be positive integers such that m>n, gcd(m,n)=1 and m \not≡ n \bmod 2. In 1956, L. Jeśmanowicz \citeJes conjectured that the equation (m² - n²)^x + (2mn)^y = (m²+n²)^z has only the positive integer solution (x,y,z) = (2,2,2). This problem is not yet solved. In this paper, combining a lower bound for linear forms in two logarithms due to M. Laurent \citeLau with some elementary methods, we prove that if mn ≡ 2 \bmod 4 and m > 30.8 n, then Jeśmanowicz' conjecture is true.

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