A note on Jeśmanowicz' conjecture concerning Pythagorean triples

Maohua Le

doi:10.1017/S0004972700033177

Abstract

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Let n be a positive integer, and let (a, b, c) be a primitive Pythagorean triple. In this paper we give certain conditions for the equation (an)x + (bn)y = (cn)z to have positive integer solutions (x, y, z) with (x, y, z) ≠ (2, 2, 2). In particular, we show that x, y and z must be distinct.

References

[1]Deng, M.-J. and Cohen, G.L., ‘On the conjecture of Jeśmanowicz concerning Pythagorean triples’, Bull. Austral. Math. Soc. 57 (1998), 515–524.CrossRef Google Scholar

[2]Jeśmanowicz, L., ‘Several remarks on Pythagorean numbers’, (in Polish), Wiadom. Mat. 1 (1955/1956), 196–202.Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

MIYAZAKI, TAKAFUMI 2009. ON THE CONJECTURE OF JEŚMANOWICZ CONCERNING PYTHAGOREAN TRIPLES. Bulletin of the Australian Mathematical Society, Vol. 80, Issue. 3, p. 413.

Miyazaki, Takafumi 2011. Jeśmanowicz' conjecture on exponential diophantine equations. Functiones et Approximatio Commentarii Mathematici, Vol. 45, Issue. 2,

YANG, ZHI-JUAN and TANG, MIN 2012. ON THE DIOPHANTINE EQUATION (8n)x+(15n)y=(17n)z. Bulletin of the Australian Mathematical Society, Vol. 86, Issue. 2, p. 348.

TANG, MIN and YANG, ZHI-JUAN 2013. JEŚMANOWICZ’ CONJECTURE REVISITED. Bulletin of the Australian Mathematical Society, Vol. 88, Issue. 3, p. 486.

Tang, Min and Weng, Jian-Xin 2014. JEŚMANOWICZ' CONJECTURE WITH FERMAT NUMBERS. Taiwanese Journal of Mathematics, Vol. 18, Issue. 3,

DENG, MOU JIE 2014. A NOTE ON THE DIOPHANTINE EQUATION . Bulletin of the Australian Mathematical Society, Vol. 89, Issue. 2, p. 316.

MA, MI-MI and WU, JIAN-DONG 2015. ON THE DIOPHANTINE EQUATION (an)x+ (bn)y= (cn)z. Bulletin of the Korean Mathematical Society, Vol. 52, Issue. 4, p. 1133.

Miyazaki, Takafumi 2015. A remark on Jeśmanowicz’ conjecture for the non-coprimality case. Acta Mathematica Sinica, English Series, Vol. 31, Issue. 8, p. 1255.

Yang, Hai and Fu, Ruiqin 2015. A note on Jeśmanowicz' conjecture concerning primitive Pythagorean triples. Journal of Number Theory, Vol. 156, Issue. , p. 183.

Han, Q. and Yuan, P. 2018. A note on Jeśmanowicz’ conjecture. Acta Mathematica Hungarica, Vol. 156, Issue. 1, p. 220.

Tingting, Wang Xiaonan, Wang and Yingzhao, Jiang 2018. An application of the Baker method to Jeśmanowicz’ conjecture on Pythagorean triples. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, Vol. 112, Issue. 2, p. 385.

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Yan, Xiao-Hui 2020. The Diophantine equation (m2 + n2)x + (2mn)y = (m + n)2z. International Journal of Number Theory, Vol. 16, Issue. 08, p. 1701.

FUJITA, YASUTSUGU and LE, MAOHUA 2021. A NOTE ON JEŚMANOWICZ’ CONJECTURE CONCERNING NONPRIMITIVE PYTHAGOREAN TRIPLES. Bulletin of the Australian Mathematical Society, Vol. 104, Issue. 1, p. 29.

Feng, Cheng and Luo, Jiagui 2022. On the exponential Diophantine equation $ \left(\frac{q^{2l}-p^{2k}}{2}n\right)^x+(p^kq^ln)^y = \left(\frac{q^{2l}+p^{2k}}{2}n\right)^z $. AIMS Mathematics, Vol. 7, Issue. 5, p. 8609.

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A note on Jeśmanowicz' conjecture concerning Pythagorean triples

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