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

Le M., SOYDAN G.

PERIODICA MATHEMATICA HUNGARICA, cilt.80, sa.1, ss.74-80, 2020 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 80 Sayı: 1
  • Basım Tarihi: 2020
  • Doi Numarası: 10.1007/s10998-019-00295-0
  • Dergi Adı: PERIODICA MATHEMATICA HUNGARICA
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET
  • Sayfa Sayıları: ss.74-80
  • Anahtar Kelimeler: Ternary purely exponential Diophantine equation, Primitive Pythagorean triple, Jesmanowicz' conjecture, Application of Baker's method
  • Bursa Uludağ Üniversitesi Adresli: Evet

Özet

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.