On the conjecture of Jesmanowicz concerning Pythagorean triples

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...

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Bibliographic Details
Published in:Bulletin of the Australian Mathematical Society 1998-06, Vol.57 (3), p.515-524
Main Authors: Deng, Moujie, Cohen, G.L.
Format: Article
Language:English
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Summary: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).
ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972700031920