An application of $ p $-adic Baker method to a special case of Jeśmanowicz' conjecture

Ziyu Dong; Zhengjun Zhao; Ziyu Dong; Zhengjun Zhao

doi:10.3934/math.2023588

AIMS Mathematics

2023, Volume 8, Issue 5: 11617-11628. doi: 10.3934/math.2023588

Previous Article Next Article

Research article

An application of $ p $-adic Baker method to a special case of Jeśmanowicz' conjecture

Ziyu Dong ,
Zhengjun Zhao ^,

School of Mathematics and Physics, Anqing Normal University, Anqing 246133, China

Received: 06 January 2023 Revised: 06 March 2023 Accepted: 12 March 2023 Published: 16 March 2023
MSC : 11D61, 11J86

In 1956, Jeśmanowicz conjectured that, for any positive integer $ n $, the Diophantine equation $ \left((f^{2}-g^{2})n\right)^{x}+\left((2fg)n\right)^{y} = \left((f^{2}+g^{2})n\right)^z $ has only the positive integral solution $ (x, y, z) = (2, 2, 2) $, where $ f $ and $ g $ are positive integers with $ f > g $, gcd$ (f, g) = 1 $, and $ f\not\equiv g\pmod {2} $. Let $ r = 6k+2 $, $ k \in \mathbb{N} $, $ k\geq25 $. In this paper, combining $ p $-adic form of Baker method with some detailed computation, we prove that if $ n $ satisfies $ n\equiv 0, 6, 9\pmod{12} $, $ f = g+1 $ and $ g = 2^{r}-1 $, then the conjecture is true.
- exponential Diophantine equation,
- Pythagorean triple,
- Jeśmanowicz' conjecture
Citation: Ziyu Dong, Zhengjun Zhao. An application of $ p $-adic Baker method to a special case of Jeśmanowicz' conjecture[J]. AIMS Mathematics, 2023, 8(5): 11617-11628. doi: 10.3934/math.2023588

Related Papers:

Abstract

In 1956, Jeśmanowicz conjectured that, for any positive integer $ n $, the Diophantine equation $ \left((f^{2}-g^{2})n\right)^{x}+\left((2fg)n\right)^{y} = \left((f^{2}+g^{2})n\right)^z $ has only the positive integral solution $ (x, y, z) = (2, 2, 2) $, where $ f $ and $ g $ are positive integers with $ f > g $, gcd$ (f, g) = 1 $, and $ f\not\equiv g\pmod {2} $. Let $ r = 6k+2 $, $ k \in \mathbb{N} $, $ k\geq25 $. In this paper, combining $ p $-adic form of Baker method with some detailed computation, we prove that if $ n $ satisfies $ n\equiv 0, 6, 9\pmod{12} $, $ f = g+1 $ and $ g = 2^{r}-1 $, then the conjecture is true.

References

[1]	M. A. Bennett, J. S. Ellenberg, N. C. Ng, The Diophantine equation $A^{4}+2^{\delta}B^{2} = C^{n}$, Int. J. Number Theory, 6 (2010), 311–338. https://dx.doi.org/10.1142/S1793042110002971 doi: 10.1142/S1793042110002971
[2]	Y. Bliu, V. Hanrot, P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math., 539 (2001), 75–122. https://dx.doi.org/10.1515/crll.2001.080 doi: 10.1515/crll.2001.080
[3]	Y. Bugeaud, Linear forms in $p$-adic logarithms and the Diophantine equation $(x^{n}-1)/(x-1) = y^{q}$, Math. Proc. Camb. Philos. Soc., 127 (1999), 373–381. https://dx.doi.org/10.1017/S0305004199003692 doi: 10.1017/S0305004199003692
[4]	V. A. Dem'janenko, On Jeśmanowicz' problem for Pythagorean numbers, Izv. Vyss. Ucebn. Zayed. Mat., 48 (1965), 52–56.
[5]	M. J. Deng, A note on the Diophantine equation $(na)^{x}+(nb)^{y} = (nc)^{z}$, Bull. Aust. Math. Soc., 89 (2014), 316–321. https://dx.doi.org/10.1017/S000497271300066X doi: 10.1017/S000497271300066X
[6]	Y. Fujita, M. H. Le, Dem'janenko's theorem on Jeśmanowicz' conjecture concerning Pythagorean triples revisited, Bull. Malays. Math. Sci. Soc., 44 (2021), 4059–4083. https://dx.doi.org/10.1007/S40840-021-01157-0 doi: 10.1007/S40840-021-01157-0
[7]	K. Györy, On the diophantine equation $n(n+1)\cdots (n+k-1) = bx^{l}$, Acta Arith., 83 (1998), 87–92. https://dx.doi.org/10.4064/aa-83-1-87-92 doi: 10.4064/aa-83-1-87-92
[8]	Y. Z. Hu, P. Z. Yuan, Jeśmanowicz' conjecture concerning Pythagorean numbers, Acta Math. Sin. Chin. Ser., 53 (2010), 297–300. https://dx.doi.org/10.12386/A2010sxxb0036 doi: 10.12386/A2010sxxb0036
[9]	L. Jeśmanowicz, Several remarks on Pythagorean numbers, Wiadom. Mat., 1 (1956), 196–202.
[10]	M. Laurent, Linear forms in two logarithms and interpolation determinants Ⅱ, Acta Arith., 133 (2008), 325–348. https://dx.doi.org/10.4064/aa133-4-3 doi: 10.4064/aa133-4-3
[11]	T. Miyazaki, Generalizations of classical results on Jeśmanowicz' conjecture concerning Pythagorean triples, J. Number Theory, 133 (2013), 583–595. https://dx.doi.org/10.1016/j.jnt.2012.08.018 doi: 10.1016/j.jnt.2012.08.018
[12]	T. Miyazaki, A remark on Jeśmanowicz' conjecture for the non-coprimality case, Acta Math. Sin. Engl. Ser., 31 (2015), 1255–1260. https://dx.doi.org/10.1007/s10114-015-4491-2 doi: 10.1007/s10114-015-4491-2
[13]	T. Miyazaki, I. Pink, Number of solutions to a special type of unit equations in two variables, arXiv, 2020. https://dx.doi.org/10.48550/arXiv.2006.15952
[14]	T. Miyazaki, I. Pink, Number of solutions to a special type of unit equations in two variables Ⅱ, arXiv, 2022. https://dx.doi.org/10.48550/arXiv.2205.11217
[15]	N. Terai, On Jeśmanowicz' conjecture concerning primitive Pythagorean triples, J. Number Theory, 141 (2014), 316–323. https://doi.org/10.1016/j.jnt.2014.02.009 doi: 10.1016/j.jnt.2014.02.009
[16]	H. Yang, R. Q. Fu, Fermat primes and Jeśmanowicz' conjecture, Adv. Math. China, 46 (2017), 857–866.
[17]	H. Yang, R. Q. Fu, A further note on Jeśmanowicz' conjecture concerning primitive pythagorean triples, Mediterr. J. Math., 19 (2022), 57–64. https://doi.org/10.1007/s00009-022-01990-y doi: 10.1007/s00009-022-01990-y

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)