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THE SHUFFLE VARIANT OF JEŚMANOWICZ’ CONJECTURE CONCERNING PYTHAGOREAN TRIPLES

Part of: Diophantine equations

Published online by Cambridge University Press:  15 August 2011

TAKAFUMI MIYAZAKI
TAKAFUMI MIYAZAKI*
Affiliation:
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan (email: miyazaki-takafumi@ed.tmu.ac.jp)
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Abstract

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Let (a,b,c) be a primitive Pythagorean triple such that b is even. In 1956, Jeśmanowicz conjectured that the equation ax+by=cz has the unique solution (x,y,z)=(2,2,2) in the positive integers. This is one of the most famous unsolved problems on Pythagorean triples. In this paper we propose a similar problem (which we call the shuffle variant of Jeśmanowicz’ problem). Our problem states that the equation cx+by=az with x,y and z positive integers has the unique solution (x,y,z)=(1,1,2) if c=b+1 and has no solutions if c>b+1 . We prove that the shuffle variant of the Jeśmanowicz problem is true if c≡1 mod b.

Keywords

exponential Diophantine equationsPythagorean tripleslower bounds for linear forms in the logarithms

MSC classification

Secondary: 11D61: Exponential equations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 3 , June 2011 , pp. 355 - 370
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

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