Let $ k, l, m_1, m_2 $ be positive integers and let both $ p $ and $ q $ be odd primes such that $ p^k = 2^{m_1}-a^{m_2} $ and $ q^l = 2^{m_1}+a^{m_2} $ where $ a $ is odd prime with $ a\equiv 5\pmod 8 $ and $ a\not\equiv 1\pmod 5 $. In this paper, using only the elementary methods of factorization, congruence methods and the quadratic reciprocity law, we show that the exponential Diophantine equation $ \left(\frac{q^{2l}-p^{2k}}{2}n\right)^x+(p^kq^ln)^y = \left(\frac{q^{2l}+p^{2k}}{2}n\right)^z $ has only the positive integer solution $ (x, y, z) = (2, 2, 2) $.
Citation: Cheng Feng, Jiagui Luo. On the exponential Diophantine equation $ \left(\frac{q^{2l}-p^{2k}}{2}n\right)^x+(p^kq^ln)^y = \left(\frac{q^{2l}+p^{2k}}{2}n\right)^z $[J]. AIMS Mathematics, 2022, 7(5): 8609-8621. doi: 10.3934/math.2022481
Let $ k, l, m_1, m_2 $ be positive integers and let both $ p $ and $ q $ be odd primes such that $ p^k = 2^{m_1}-a^{m_2} $ and $ q^l = 2^{m_1}+a^{m_2} $ where $ a $ is odd prime with $ a\equiv 5\pmod 8 $ and $ a\not\equiv 1\pmod 5 $. In this paper, using only the elementary methods of factorization, congruence methods and the quadratic reciprocity law, we show that the exponential Diophantine equation $ \left(\frac{q^{2l}-p^{2k}}{2}n\right)^x+(p^kq^ln)^y = \left(\frac{q^{2l}+p^{2k}}{2}n\right)^z $ has only the positive integer solution $ (x, y, z) = (2, 2, 2) $.
| [1] | M. J. Deng, G. L. Cohen, On the conjecture of Jeśmanowicz concerning Pythagorean triples, Bull. Austral. Math. Soc., 57 (1998), 515–524. |
| [2] |
M. J. Deng, A note on the Diophantine equation $(an)^x +(bn)^y = (cn)^z$, Bull. Aust. Math. Soc., 89 (2014), 316–321. https://doi.org/10.1017/S000497271300066X doi: 10.1017/S000497271300066X
|
| [3] | N. Deng, P. Z. Yuan, W. Luo, Number of solutions to $ka^x+lb^y = c^z$, J. Number Theory, 187 (2018), 250–263. |
| [4] |
Y. Z. Hu, M. H. Le, An upper bound for the number of solutions of tenary purely exponential Diophantine equations, J. Number Theory, 187 (2018), 62–73. https://doi.org/10.1016/j.jnt.2017.07.004 doi: 10.1016/j.jnt.2017.07.004
|
| [5] | L. Jeśmanowicz, Several remarks on Pythagorean numbers, Wiadom. Mat., 1 (1955), 196–202. |
| [6] |
M. H. Le, A note on Jeśmanowicz' conjecture concerning Pythagorean triples, Bull. Austral. Math. Soc., 59 (1999), 477–480. https://doi.org/10.1017/S0004972700033177 doi: 10.1017/S0004972700033177
|
| [7] |
M. M. Ma, Y. G. Chen, Jeśmanowicz' conjecture on Pythagorean triples, Bull. Austral. Math. Soc., 96 (2017), 30–35. https://doi.org/10.1017/S0004972717000107 doi: 10.1017/S0004972717000107
|
| [8] |
T. Miyazaki, Generalizations of classical results on Jeśmanowicz' conjecture concerning Pythagorean triples, J. Number Theory, 133 (2013), 583–595. https://doi.org/10.1016/j.jnt.2012.08.018 doi: 10.1016/j.jnt.2012.08.018
|
| [9] |
T. Miyazaki, P. Z. Yuan, D. Wu, Generalizations of classical results on Jeśmanowicz' conjecture concerning Pythagorean triples II, J. Number Theory, 141 (2014), 184–201. https://doi.org/10.1016/j.jnt.2014.01.011 doi: 10.1016/j.jnt.2014.01.011
|
| [10] |
T. Miyazaki, A remark on Jeśmanowicz' conjecture for non-coprimality case, Acta Math. Sin.-English Ser., 31 (2015), 1225–1260. https://doi.org/10.1007/s10114-015-4491-2 doi: 10.1007/s10114-015-4491-2
|
| [11] | W. Sierpinski, On the equation $3^x +4^y = 5^z$, Wiadom. Mat., 1 (1955/1956), 194–195. |
| [12] |
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
|
| [13] |
M. Tang, J. X. Weng, Jeśmanowicz' conjecture with Fermat numbers, Taiwanese J. Math., 18 (2014), 925–930. https://doi.org/10.11650/tjm.18.2014.3942 doi: 10.11650/tjm.18.2014.3942
|
| [14] |
Z. J. Yang, M. Tang, On the Diophantine equation $(8n)^x + (15n)^y = (17n)^z$, Bull. Austral. Math. Soc., 86 (2010), 348–352. https://doi.org/10.1017/S000497271100342X doi: 10.1017/S000497271100342X
|
| [15] |
Z. J. Yang, M. Tang, On the Diophantine equation $(8n)^x +(15n)^y = (17n)^z$, Bull. Austral. Math. Soc., 86 (2012), 348–352. https://doi.org/10.1017/S000497271100342X doi: 10.1017/S000497271100342X
|
| [16] | P. Z. Yuan, Q. Han, Jeśmanowicz conjectuee and related questions, Acta Arith., 184 (2018), 37–49. |
Cheng Feng, Jiagui Luo. On the exponential Diophantine equation $ \left(\frac{q^{2l}-p^{2k}}{2}n\right)^x+(p^kq^ln)^y = \left(\frac{q^{2l}+p^{2k}}{2}n\right)^z $[J]. AIMS Mathematics, 2022, 7(5): 8609-8621. doi: 10.3934/math.2022481
DownLoad: