Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-29T11:58:45.409Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Deng, M. and Cohen, G. L., ‘On the conjecture of Jeśmanowicz concerning Pythagorean triples’, Bull. Aust. Math. Soc. 57 (1998), 515–524.Google Scholar
[2]Jeśmanowicz, L., ‘Several remarks on Pythagorean numbers’, Wiadom. Mat. 1 (1955/56), 196–202.Google Scholar
[3]Maohua, L., ‘A note on Jeśmanowicz’ conjecture’, Colloq. Math. 69 (1995), 47–51.Google Scholar
[4]Maohua, L., ‘On Jeśmanowicz’ conjecture concerning Pythagorean triples’, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 97–98.Google Scholar
[5]Maohua, L., ‘A note on Jeśmanowicz’ conjecture concerning Pythagorean triples’, Bull. Aust. Math. Soc. 59 (1999), 477–480.Google Scholar
[6]Maohua, L., ‘A note on Jeśmanowicz’ conjecture concerning primitive Pythagorean triples’, Acta Arith. 138 (2009), 137–144.Google Scholar
[7]Sierpiński, W., ‘On the equation 3x+4y=5z’, Wiadom. Mat. 1 (1955/56), 194–195.Google Scholar
[8]Takakuwa, K., ‘A remark on Jeśmanowicz’ conjecture’, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), 109–110.Google Scholar
[9]Wenduan, L., ‘On the Pythagorean numbers 4n 2−1,4n and 4n 2+1’, Acta Sci. Natur. Univ. Szechuan 2 (1959), 39–42.Google Scholar