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Published online by Cambridge University Press:  15 August 2011

Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan (email:
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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.

MSC classification

Research Article
Copyright © Australian Mathematical Publishing Association Inc. 2011


[1]Cao, Z. F., ‘A note on the Diophantine equation a x+b y=c z’, Acta Arith. 91 (1999), 8593.Google Scholar
[2]Dem’janenko, V. A., ‘On Jeśmanowicz’ problem for Pythagorean numbers’, Izv. Vyssh. Uchebn. Zaved. Mat. 48 (1965), 5256 (in Russian).Google Scholar
[3]Deng, M.-J. and Cohen, G. L., ‘A note on a conjecture of Jeśmanowicz’, Colloq. Math. 86 (2000), 2530.Google Scholar
[4]He, B. and Togbé, A., ‘The Diophantine equation n x+(n+1)y=(n+2)z revisited’, Glasg. Math. J. 51 (2009), 659667.Google Scholar
[5]Jeśmanowicz, L., ‘Several remarks on Pythagorean numbers’, Wiadom. Mat. 1 (1955/56), 196202 (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), 3942 (in Chinese).Google Scholar
[7]Mahler, K., ‘Zur Approximation algebraischer Zahlen I: Über den grössten Primteiler binärer Formen’, Math. Ann. 107 (1933), 691730.Google Scholar
[8]Mignotte, M., ‘A corollary to a theorem of Laurent–Mignotte–Nesterenko’, Acta Arith. 86 (1998), 101111.Google 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), 413422.Google Scholar
[12]Nagell, T., ‘Sur une classe d’equations exponentielles’, Ark. Mat. 3 (1958), 569582.Google Scholar
[13]Scott, R., ‘On the equations p xb y=c and a x+b y=c z’, J. Number Theory 44 (1993), 153165.Google Scholar
[14]Sierpiński, W., ‘On the equation 3x+4y=5z’, Wiadom. Mat. 1 (1955/56), 194195 (in Polish).Google Scholar