Abstract
Let (a, b, c) be a primitive Pythagorean triple satisfying \(a^2+b^2=c^2\). In 1956, Jeśmanowicz conjectured that the exponential Diophantine equation \((na)^x+(nb)^y=(nc)^z\) has no positive integer solutions other than \((x,y,z)=(2,2,2)\) for any positive integer n. In this paper, we obtain an effective sufficient condition for the conjecture for \((a,b,c)=(4k^2-1,4k,4k^2+1)\) with \(k=p^{\alpha }\) where \(p=4m+3\) is a prime number and \(\alpha \) is a positive integer. In addition, numerical results are included, which say that, for example, if \(\alpha =1\), then the conjecture is true for \(0\le m\le 10^7\). If we assume a certain conjecture related to abc conjecture, we can prove completely Jeśmanowicz’ conjecture of the above form for any \(\alpha \) and m.
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References
M. Bennett, C. Skinner, Ternary Diophantine equations via Galois representations and modular forms. Canad. J. Math. 56, 23–54 (2004)
M. Bennett, V. Vatsal, S. Yazdani, Ternary Diophantine equations of signature \((p, p,3)\). Compos. Math. 140, 1399–1416 (2004)
H. Cohen, Number theory. Vol. II. Analytic and modern tools, In: Graduate Texts in Mathematics, vol. 240, (Springer, New York, 2007), pp. 596
M. Deng, A note on the Diophantine equation \((na)^x+(nb)^y=(nc)^z\). Bull. Aust. Math. Soc. 89, 316–321 (2014)
R. Fu, H. Yang, A note on the exceptional solutions of Jeśmanowicz’ conjecture concerning primitive Pythagorean triples. Period. Math. Hungar. 81, 275–283 (2020)
Y. Fujita, M. Le, A note on Jeśmanowicz’ conjecture concerning nonprimitive Pythagorean triples. Bull. Aust. Math. Soc. 104, 29–39 (2021)
A. Granville, T.J. Tucker, It’s as easy as \(abc\). Not. Amer. Math. Soc. 49, 1224–1231 (2002)
L. Jeśmanowicz, Several remarks on Pythagorean numbers. Wiadom. Mat. 1, 196–202 (1955/1956)
S. Laishram, T.N. Shorey, Baker’s explicit \(abc\)-conjecture and applications. Acta Arith. 155, 419–429 (2012)
M. Le, R. Scott, R. Styer, A survey on the ternary purely exponential Diophantine equation \(a^x+b^y=c^z\). Surv. Math. Appl. 14, 109–140 (2019)
M. Le, G. Soydan, An application of Baker’s method to the Jeśmanowicz’ conjecture on primitive Pythagorean triples. Period. Math. Hungar. 80, 74–80 (2020)
W. Lu, On the Pythagorean numbers \(4n^2-1\), \(4n\) and \(4n^2+1\). Acta Sci. Natur. Univ. Szechuan 2, 39–42 (1959)
M. Ma, J. Wu, On the Diophantine equation \((an)^x+(bn)^y=(cn)^z\). Bull. Korean Math. Soc. 52, 1133–1138 (2015)
T. Miyazaki, A remark on Jeśmanowicz’ conjecture for the non-coprimality case. Acta Math. Sin. Engl. Ser. 31, 1255–1260 (2015)
T. Miyazaki, Contributions to some conjectures on a ternary exponential Diophantine equation. Acta Arith. 186, 1–36 (2018)
C. Sun, Z. Cheng, On Jeśmanowicz’ conjecture concerning Pythagorean triples. J. Math. Res. Appl. 35(2), 143–148 (2015)
N. Terai, On Jeśmanowicz’ conjecture concerning primitive Pythagorean triples. J. Number Theory 141, 316–323 (2014)
Acknowledgements
The first named author is supported by JSPS KAKENHI Grant Number JP19K03447. The authors thank the referee for careful reading and valuable suggestions.
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Kitayama, H., Tagawa, H. & Urahashi, K. Jeśmanowicz’ conjecture for non-primitive Pythagorean triples. Period Math Hung 86, 442–453 (2023). https://doi.org/10.1007/s10998-022-00482-6
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DOI: https://doi.org/10.1007/s10998-022-00482-6