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JEŚMANOWICZ’ CONJECTURE REVISITED

Part of: Diophantine equations

Published online by Cambridge University Press:  15 February 2013

MIN TANG  and
ZHI-JUAN YANG
MIN TANG*
Affiliation:
School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, PR China
ZHI-JUAN YANG
Affiliation:
School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, PR China
*
✉[email protected]
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Abstract

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Let $a, b, c$ be relatively prime positive integers such that ${a}^{2} + {b}^{2} = {c}^{2} $. In 1956, Jeśmanowicz conjectured that for any positive integer $n$, the only solution of $\mathop{(an)}\nolimits ^{x} + \mathop{(bn)}\nolimits ^{y} = \mathop{(cn)}\nolimits ^{z} $ in positive integers is $(x, y, z)= (2, 2, 2)$. In this paper, we consider Jeśmanowicz’ conjecture for Pythagorean triples $(a, b, c)$ if $a= c- 2$ and $c$ is a Fermat prime. For example, we show that Jeśmanowicz’ conjecture is true for $(a, b, c)= (3, 4, 5)$, $(15, 8, 17)$, $(255, 32, 257)$, $(65535, 512, 65537)$.

Keywords

Jeśmanowicz’ conjectureDiophantine equation

MSC classification

Secondary: 11D61: Exponential equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 3 , December 2013 , pp. 486 - 491
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

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