An application of $ p $-adic Baker method to a special case of Jeśmanowicz' conjecture

Abstract

<abstract><p>In 1956, Jeśmanowicz conjectured that, for any positive integer $ n $, the Diophantine equation $ \left((f^{2}-g^{2})n\right)^{x}+\left((2fg)n\right)^{y} = \left((f^{2}+g^{2})n\right)^z $ has only the positive integral solution $ (x, y, z) = (2, 2, 2) $, where $ f $ and $ g $ are positive integers with $ f &gt; g $, gcd$ (f, g) = 1 $, and $ f\not\equiv g\pmod {2} $. Let $ r = 6k+2 $, $ k \in \mathbb{N} $, $ k\geq25 $. In this paper, combining $ p $-adic form of Baker method with some detailed computation, we prove that if $ n $ satisfies $ n\equiv 0, 6, 9\pmod{12} $, $ f = g+1 $ and $ g = 2^{r}-1 $, then the conjecture is true.</p></abstract>