On the conjecture of Jesmanowicz concerning Pythagorean triples

Abstract

Let a, b, c be relatively prime positive integers such that a 2 + b 2 = c 2 .Jesmanowicz conjectured in 1956 that for any given positive integer n the only solution of (an) x + (bn) v = (en)* in positive integers is x -y = z = 2. Building on the work of earlier writers for the case when n = 1 and c = b + 1, we prove the conjecture when n > l , c = b + l and certain further divisibility conditions are satisfied. This leads to the proof of the full conjecture for the five triples

A note on Jeśmanowicz' conjecture concerning primitive Pythagorean triples

“…By the results of [4] and [6], the theorem holds for 1 ≤ r ≤ 3. It suffices to prove the theorem for r ≥ 4.…”

“…Deng and G.L. Cohn [6] proved that if r ∈ {1, 2}, then (3) has no exceptional solutions. M.J. Deng [4] proved the same result for r = 3.…”

A remark on Jeśmanowicz’ conjecture for the non-coprimality case

“…For some results in this direction, we refer to the papers of Deng and Cohen [3], Yang and Tang [14], Tang and Yang [11], Deng [2], Tang and Weng [12], and the references given there. Several of these concern the case where b is a power of 2.…”

ON THE DIOPHANTINE EQUATION (an)x+ (bn)y= (cn)z

“…When n > 1, only a few results on this conjecture are known. For any positive integer t with t > 1, let P (t) denote the product of distinct prime factors of t. In 1998, Deng and Cohen [3] proved that if n > 1, c = b + 1, a is a prime power and either P (b) | n or P (n) ∤ b, then (1.1) has only the solution (x, y, z) = (2, 2, 2). In 1999, Le [7] gave certain necessary conditions for (1.2) to have positive integer solutions (x, y, z) with (x, y, z) = (2, 2, 2).…”