Let $ k, l, m_1 $ and $ 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 a positive integer with $ a\equiv {\pm 3}\pmod 8 $. In this paper, using only the elementary methods of factorization, congruence methods and the quadratic reciprocity law, we show that Je$ \acute{s} $manowicz' a conjecture holds for the following set of primitive Pythagorean numbers:
$ \frac{q^{2l}-p^{2k}}{2}, p^kq^l, \frac{q^{2l}+p^{2k}}{2}. $
We also prove that Je$ \acute{s} $manowicz' conjecture holds for non-primitive Pythagorean numbers:
$ n\frac{q^{2l}-p^{2k}}{2}, np^kq^l, n\frac{q^{2l}+p^{2k}}{2}, $
for any positive integer $ n $ if for $ a = a_1a_2 $ with $ a_1\equiv 1 \pmod 8 $ not a square and $ \gcd(a_1, a_2) = 1 $, then there exists a prime divisor $ P $ of $ a_2 $ such that $ \left(\frac{a_1}{P}\right) = -1 $ and $ 2|m_1, a\equiv 5 \pmod 8 $ or $ 2\not|m_2, a\equiv 3\pmod 8 $.
Citation: Nan Fan, Jiagui Luo. On the conjecture of Je$ \acute{\textbf{s}} $manowicz[J]. AIMS Mathematics, 2023, 8(6): 14232-14252. doi: 10.3934/math.2023728
Let $ k, l, m_1 $ and $ 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 a positive integer with $ a\equiv {\pm 3}\pmod 8 $. In this paper, using only the elementary methods of factorization, congruence methods and the quadratic reciprocity law, we show that Je$ \acute{s} $manowicz' a conjecture holds for the following set of primitive Pythagorean numbers:
$ \frac{q^{2l}-p^{2k}}{2}, p^kq^l, \frac{q^{2l}+p^{2k}}{2}. $
We also prove that Je$ \acute{s} $manowicz' conjecture holds for non-primitive Pythagorean numbers:
$ n\frac{q^{2l}-p^{2k}}{2}, np^kq^l, n\frac{q^{2l}+p^{2k}}{2}, $
for any positive integer $ n $ if for $ a = a_1a_2 $ with $ a_1\equiv 1 \pmod 8 $ not a square and $ \gcd(a_1, a_2) = 1 $, then there exists a prime divisor $ P $ of $ a_2 $ such that $ \left(\frac{a_1}{P}\right) = -1 $ and $ 2|m_1, a\equiv 5 \pmod 8 $ or $ 2\not|m_2, a\equiv 3\pmod 8 $.
| [1] | Sierpin$\acute{s}$ki, On the equation $3^x +4^y = 5^z$, Wiadom. Math., 1 (1955/1956), 194–195. |
| [2] | L. Je$\acute{s}$manowicz, Several remarks on Pythagorean numbers, Wiadom. Mat., 1 (1955/1956), 196–202. |
| [3] | C. Ke, On Pythagorean numbers, J. Sichuan Univ. Nat. Sci., 1 (1958), 73–80. |
| [4] | T. J$\acute{o}$zefiak, On a hypothesis of Je$\acute{s}$manowicz L. concerning Pythagorean numbers, Prace. Math., 5 (1961), 119–123. |
| [5] | V. A. Dem'janenko, On Ye$\acute{s}$manowicz' problem for Pythagorean numbers, Izv. Vyssh. Uchebn. Zaved. Mat., 5 (1965), 52–56. |
| [6] | A. Grytczuk, A. Grelak, On the equation $a^x+b^y = c^z$, Comment. Math., 24 (1984), 269–275. |
| [7] |
K. Takakuwa, Y. Asaeda, On a conjecture on Pythagorean numbers, Proc. Japan Acad. Ser. A. Math. Sci., 69 (1993), 252–255. http://doi.org/10.3792/pjaa.69.252 doi: 10.3792/pjaa.69.252
|
| [8] | M. J. Deng, G. L. Cohen, On the conjecture of Je$\acute{s}$manowicz concerning Pythagorean triples, Bull. Aust. Math. Soc., 57 (1998), 515–524. |
| [9] | M. H. Le, A note on Je$\acute{s}$manowicz conjecture, Bull. Aust. Math. Soc., 59 (1999), 477–480. |
| [10] |
H. Yang, R. Q. Fu, A note on Je$\acute{s}$manowicz conjecture concerning primitive Pythagorean triples, J. Number Theory, 156 (2015), 183–194. https://doi.org/10.1016/j.jnt.2015.04.009 doi: 10.1016/j.jnt.2015.04.009
|
| [11] |
M. Tang, Z. J. Yang, Je$\acute{s}$manowicz conjecture revisited, Bull. Aust. Math. Soc., 88 (2013), 486–491. http://doi.org/10.1017/S0004972713000038 doi: 10.1017/S0004972713000038
|
| [12] |
M. Tang, J. X. Weng, Je$\acute{s}$manowicz' conjecture with Fermat numbers, Taiwanese J. Math., 18 (2014), 925–930. http://doi.org/10.11650/tjm.18.2014.3942 doi: 10.11650/tjm.18.2014.3942
|
| [13] | X. W. Zhang, W. P. Zhang, The exponential Diophantine equation $((2^2m-1)n)^x+(2^{m+1}n)^y = ((2^2m + 1)n)^z$, Bull. Math. Soc. Sci. Math. Roumanie, 57 (2014), 337–344. |
| [14] |
Z. J. Yang, M. Tang, On the Diophantine equation $(8n)^x + (15n)^y = (17n)^z$, Bull. Aust. Math. Soc., 86 (2012), 348–352. http://doi.org/10.1017/S000497271100342X doi: 10.1017/S000497271100342X
|
| [15] |
M. J. Deng, A note on the Diophantine equation $(na)^x +(nb)^y = (nc)^z$, Bull. Aust. Math. Soc., 89 (2014), 316–321. http://doi.org/10.1017/S000497271300066X doi: 10.1017/S000497271300066X
|
| [16] |
M. M. Ma, J. D. Wu, On the Diophantine equation $(an)^x +(bn)^y = (cn)^z$, Bull. Korean Math. Soc., 52 (2015), 1133–1138. http://doi.org/10.4134/BKMS.2015.52.4.1133 doi: 10.4134/BKMS.2015.52.4.1133
|
| [17] |
T. Miyazaki, A remark on Je$\acute{s}$manowicz conjecture for non-coprimality case, Acta Math. Sin. (Engl. Ser.), 31 (2015), 1255–1260. http://doi.org/10.1007/s10114-015-4491-2 doi: 10.1007/s10114-015-4491-2
|
| [18] |
G. Soydan, M. Demirci, I. N. Cangul, A. Togb$\acute{e}$, On the conjecture of Je$\acute{s}$manowicz, Int. J. Appl. Math., 56 (2017), 46–72. http://doi.org/10.48550/arXiv.1706.05480 doi: 10.48550/arXiv.1706.05480
|
| [19] |
C. Feng, J. G. Luo, On the Diophantine equation $\left(\frac{q^2l-p^2k}{2}n\right)^x+(q^lp^kn)^y = \left(\frac{q^2l+p^2k}{2}n\right)^z$, AIMS Mathematics, 7 (2022), 8609–8621. http://doi.org/10.3934/math.2022481 doi: 10.3934/math.2022481
|
| [20] |
N. J. Deng, P. Z. Yuan, W. Y. Luo, Number of solutions to $ka^x+lb^y = c^z$, J. Number Theory, 187 (2018), 250–263. http://doi.org/10.1016/j.jnt.2017.10.031 doi: 10.1016/j.jnt.2017.10.031
|
| [21] |
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. http://dx.doi.org/10.1016/j.jnt.2017.07.004 doi: 10.1016/j.jnt.2017.07.004
|
| [22] |
T. Miyazaki, Generalizations of classical results on Je$\acute{s}$manowicz' conjecture concerning Pythagorean triples, J. Number Theory, 133 (2013), 583–595. http://doi.org/10.1016/j.jnt.2012.08.018 doi: 10.1016/j.jnt.2012.08.018
|
| [23] |
T. Miyazaki, P. Z. Yuan, D. Y. Wu, Generalizations of classical results on Je$\acute{s}$manowicz' conjecture concerning Pythagorean triples Ⅱ, J. Number Theory, 141 (2014), 184–201. http://dx.doi.org/10.1016/j.jnt.2014.01.011 doi: 10.1016/j.jnt.2014.01.011
|
| [24] |
N. Terai, On Je$\acute{s}$manowicz' conjecture concerning primitive Pythagorean triples, J. Number Theory, 141 (2014), 316–323. http://doi.org/10.1016/j.jnt.2014.02.009 doi: 10.1016/j.jnt.2014.02.009
|
| [25] |
P. Z. Yuan, Q. Han, Je$\acute{s}$manowicz conjecture and related questions, Acta Arith., 184 (2018), 37–49. http://doi.org/10.4064/aa170508-17-9 doi: 10.4064/aa170508-17-9
|
Nan Fan, Jiagui Luo. On the conjecture of Je$ \acute{\textbf{s}} $manowicz[J]. AIMS Mathematics, 2023, 8(6): 14232-14252. doi: 10.3934/math.2023728
DownLoad: