Abstract
Let (a, b, c) be pairwise relatively prime integers such that \(a^2 + b^2 = c^2\) . In 1956, Jeśmanowicz conjectured that the only solution of \(a^x + b^y = c^z\) in positive integers is \((x,y,z)=(2,2,2)\). In this note we prove a polynomial analogue of this conjecture.
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References
V.A. Dem’janenko, On Jeśmanowicz’ problem for Pythagorean numbers. Izv. Vyssh. Uchebn. Zaved. Mat. 48, 52–56 (1965) (in Russian)
M.-J. Deng, D.-M. Huang, A note on Jeśmanowicz’ conjecture concerning primitive Pythagorean triples. Bull. Aust. Math. Soc. 95(1), 5–13 (2017)
Y. Fujita, T. Miyazaki, Jeśmanowicz’ conjecture with congruence relations. Colloq. Math. 128, 211–222 (2012)
L. Jes̀manowicz, Several remarks on Pythagorean numbers. Wiadom. Math. 1, 196–202 (1955/56)
C. Ko, On Pythagorean numbers. J. Sichuan Univ. Nat. Sci. 1, 73–80 (1958) (in Chinese)
C. Ko, On Jes̀manowicz’ conjecture. J. Sichuan Univ. Nat. Sci. 2, 81–90 (1958) (in Chinese)
K.K. Kubota, Pythagorean triples in unique factorization domains. Am. Math. Mon. 79(5), 503–505 (1972)
M. Le, A note on Jes̀manowicz’ conjecture. Colloq. Math. 69, 47–51 (1995)
W.D. Lu, On the Pythagorean numbers \(4n^2-1\), \(4n\) and \(4n^2+1\). Acta Sci. Nat. Univ. Szechuan. 2, 39–42 (1959)
M.-M. Ma, Y.C. Chen, Jes̀manowicz conjecture on Pythagorean triples. Bull. Aust. Math. Soc. 96(1), 30–35 (2017)
T. Miyazaki, Jes̀manowicz conjecture on exponential Diophantine equations. Funct. Approx. Comment. Math. 45, 207–229 (2011)
T. Miyazaki, P. Yuan, D. Wu, Generalizations of classical results on Jes̀manowicz conjecture concerning Pythagorean triples II. J. Number Theory 141, 184–201 (2014)
V.D. Podsypanin, On a property of Pythagorean numbers. Izv. Vyssh. Uchebn. Zaved. Math. 4, 130–133 (1962) (in Russian)
W. Sierpinski, On the equation \(3^x + 4^y = 5^z\). Wiadom. Math. 1, 194–195 (1955/56)
N. Snyder, An alternate proof of Mason’s theorem. Elem. Math. 55, 93–94 (2000)
M. Tang, J.-X. Weng, Jes̀manowicz’ conjecture with Fermat numbers. Taiwan. J. Math. 18(3), 925–930 (2014)
M. Tang, Z.J. Yang, Jes̀manowicz’ conjecture revisited. Bull. Aust. Math. Soc. 88, 486–491 (2013)
N. Terai, On Jeśmanowicz’ conjecture concerning primitve Pythagorean triples. J. Number Theory 141, 316–323 (2014)
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Dimabayao, J.T. Jeśmanowicz’ conjecture for polynomials. Period Math Hung 82, 29–38 (2021). https://doi.org/10.1007/s10998-020-00339-w
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DOI: https://doi.org/10.1007/s10998-020-00339-w