On the conjecture of Jesmanowicz concerning Pythagorean triples

Author:

Deng Moujie,Cohen G.L.

Abstract

Let a, b, c be relatively prime positive integers such that a2 + b2 = c2. Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of (an)x + (bn)y = (en)z 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 > 1, c = b + 1 and certain further divisibility conditions are satisfied. This leads to the proof of the full conjecture for the five triples (a, b, c) = (3, 4, 5), (5, 12, 13), (7, 24, 25), (9, 40, 41) and (11, 60, 61).

Publisher

Cambridge University Press (CUP)

Subject

General Mathematics

Reference9 articles.

1. On Jeśmanowicz' conjecture concerning Pythagorean numbers;Maohua;Proc. Japan Acad. Ser. A Math. Sci.,1996

2. Several remarks on Pythagorean numbers;Jeśmanowicz;Wiadom. Mat.,1955

3. On Jeśmanowicz' problem for Pythagorean numbers;Dem'janenko;Izv. Vysšh. Učebn. Zaved. Mat.,1965

4. A remark on Jeśmanowicz' conjecture

5. A note on Jeśmanowicz' conjecture;Maohua;Colloq. Math.,1995

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