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


Le M., SOYDAN G.

PERIODICA MATHEMATICA HUNGARICA, vol.80, no.1, pp.74-80, 2020 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 80 Issue: 1
  • Publication Date: 2020
  • Doi Number: 10.1007/s10998-019-00295-0
  • Journal Name: PERIODICA MATHEMATICA HUNGARICA
  • Journal Indexes: Science Citation Index Expanded, Scopus, MathSciNet, zbMATH, DIALNET
  • Page Numbers: pp.74-80
  • Keywords: Ternary purely exponential Diophantine equation, Primitive Pythagorean triple, Jesmanowicz' conjecture, Application of Baker's method

Abstract

Let m, n be positive integers such that m > n, gcd(m,n) = 1 and m not equivalent to n(mod2) . In 1956, L. Jesmanowicz conjectured that the equation (m(2)-n(2))(x) + (2mn)(y) = (m(2) + n(2))(z) has only the positive integer solution (x,y,z)=(2,2,2). This conjecture is still unsolved. In this paper, combining a lower bound for linear forms in two logarithms due to M. Laurent with some elementary methods, we prove that if mn equivalent to 2(mod 4) and m > 30.8n, then Jesmanowicz' conjecture is true.