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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

References

[1]Cao, Z. F., ‘A note on the Diophantine equation a x+b y=c z’, Acta Arith. 91 (1999), 85–93.CrossRefGoogle Scholar
[2]Dem’janenko, V. A., ‘On Jeśmanowicz’ problem for Pythagorean numbers’, Izv. Vyssh. Uchebn. Zaved. Mat. 48 (1965), 52–56 (in Russian).Google Scholar
[3]Deng, M.-J. and Cohen, G. L., ‘A note on a conjecture of Jeśmanowicz’, Colloq. Math. 86 (2000), 25–30.CrossRefGoogle Scholar
[4]He, B. and Togbé, A., ‘The Diophantine equation n x+(n+1)y=(n+2)z revisited’, Glasg. Math. J. 51 (2009), 659–667.CrossRefGoogle Scholar
[5]Jeśmanowicz, L., ‘Several remarks on Pythagorean numbers’, Wiadom. Mat. 1 (1955/56), 196–202 (in Polish).Google Scholar
[6]Lu, W. T., ‘On the Pythagorean numbers 4n 2−1,4n and 4n 2+1’, Acta Sci. Natur. Univ. Szechuan 2 (1959), 39–42 (in Chinese).Google Scholar
[7]Mahler, K., ‘Zur Approximation algebraischer Zahlen I: Über den grössten Primteiler binärer Formen’, Math. Ann. 107 (1933), 691–730.CrossRefGoogle Scholar
[8]Mignotte, M., ‘A corollary to a theorem of Laurent–Mignotte–Nesterenko’, Acta Arith. 86 (1998), 101–111.CrossRefGoogle Scholar
[9]Miyazaki, T., ‘Jeśmanowicz’ conjecture on exponential Diophantine equations’, Funct. Approx. Comment. Math., in press.Google Scholar
[10]Miyazaki, T., ‘Generalizations of classical results on Jeśmanowicz’ conjecture concerning Pythagorean triples’, Diophantine Analysis and Related Fields 2010, AIP Conf. Proc., Vol. 1264, Tokyo, Japan, 2010 (American Institute of Physics, Melville, NY, 2010), pp. 41–51.Google Scholar
[11]Miyazaki, T., ‘On the conjecture of Jeśmanowicz concerning Pythagorean triples’, Bull. Aust. Math. Soc. 80 (2009), 413–422.CrossRefGoogle Scholar
[12]Nagell, T., ‘Sur une classe d’equations exponentielles’, Ark. Mat. 3 (1958), 569–582.CrossRefGoogle Scholar
[13]Scott, R., ‘On the equations p x−b y=c and a x+b y=c z’, J. Number Theory 44 (1993), 153–165.CrossRefGoogle Scholar
[14]Sierpiński, W., ‘On the equation 3x+4y=5z’, Wiadom. Mat. 1 (1955/56), 194–195 (in Polish).Google Scholar