Generalizations of classical results on Jeśmanowiczʼ conjecture concerning Pythagorean triples

In 1956 L. Jeśmanowicz conjectured, for any primitive Pythagorean triple (a,b,c) satisfying a2+b2=c2, that the equation ax+by=cz has the unique solution (x,y,z)=(2,2,2) in positive integers x, y and z. This is a famous unsolved problem on Pythagorean numbers. In this paper we broadly extend many of...

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Bibliographic Details
Published inJournal of number theory Vol. 133; no. 2; pp. 583 - 595
Main Author Miyazaki, Takafumi
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.02.2013
Subjects
Exponential Diophantine equations
Jeśmanowiczʼ conjecture
Pythagorean triples
secondary
Exponential Diophantine equations
Pythagorean triples
Jeśmanowiczʼ conjecture
primary
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Summary:In 1956 L. Jeśmanowicz conjectured, for any primitive Pythagorean triple (a,b,c) satisfying a2+b2=c2, that the equation ax+by=cz has the unique solution (x,y,z)=(2,2,2) in positive integers x, y and z. This is a famous unsolved problem on Pythagorean numbers. In this paper we broadly extend many of classical well-known results on the conjecture. As a corollary we can verify that the conjecture is true if a−b=±1.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2012.08.018