A Finite Verification of Jeśmanowicz' Conjecture under 2 ∥ mn for n ≤ 200

Abstract

Jeśmanowicz' conjecture is a long-standing problem on exponential Diophantine equations associated with primitive Pythagorean triples. Let m > n be coprime positive integers of opposite parity. The conjecture asserts that (m² − n²)^x + (2mn)^y = (m² + n²)^z has only the positive integer solution (x, y, z) = (2, 2, 2). In this article, we prove the conjecture in the case 2 ∥ mn for n ≤ 200. The proof combines four known results with an exact finite modular argument. The known results reduce the problem to 170 admissible parameter pairs, and every exceptional solution for a remaining pair must satisfy y = 1. All residual cases are then excluded by comparing finite power-residue sets modulo a fixed collection of 13 moduli. The verification uses only exact integer and modular arithmetic and is reproducible from the script provided in the appendix.