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ON THE DIOPHANTINE EQUATION (8n)x+(15n)y=(17n)z

Part of: Diophantine equations

Published online by Cambridge University Press:  07 February 2012

ZHI-JUAN YANG  and
MIN TANG
ZHI-JUAN YANG
Affiliation:
Department of Mathematics, Anhui Normal University, Wuhu 241000, China
MIN TANG*
Affiliation:
Department of Mathematics, Anhui Normal University, Wuhu 241000, China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let a,b,c be relatively prime positive integers such that a2+b2=c2. Half a century ago, Jeśmanowicz [‘Several remarks on Pythagorean numbers’, Wiadom. Mat.1 (1955/56), 196–202] conjectured that for any given positive integer n the only solution of (an)x+(bn)y=(cn)z in positive integers is (x,y,z)=(2,2,2). In this paper, we show that (8n)x+(15n)y=(17n)z has no solution in positive integers other than (x,y,z)=(2,2,2).

Keywords

Jeśmanowicz’s conjectureDiophantine equation

MSC classification

Secondary: 11D61: Exponential equations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 2 , October 2012 , pp. 348 - 352
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

This work was supported by the National Natural Science Foundation of China, Grant No 10901002.

References

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