Abstract
Let $f,g$ be positive integers such that $f>g$, $\gcd(f,g)=1$ and $f\not\equiv g \pmod{2}$. In 1993, N. Terai conjectured that the equation $x^2+(f^2-g^2)^y=(f^2+g^2)^z$ has only one positive integer solution $(x,y,z)=(2fg,2,2)$. This is a problem that has not been solved yet. In this paper, using elementary number theory methods with some known results on higher Diophantine equations, we prove that if $f=2^rs$ and $g=1$, where $r,s$ are positive integers satisfying $2\nmid s$, $r\ge 2$ and $s<2^{r-1}$, then Terai's conjecture is true.
Keywords
polynomial-exponential Diophantine equation generalized Ramanujan-Nagell equation primitive Pythagorean triple
References
- [1] M.A. Bennett, J.S. Ellenberg and N.C. Ng, The Diophantine equation $A^4+2^\delta B^2=C^n$, Int. J. Number Theory, 6 (2), 311–338, 2010.
- [2] X.G. Chen and M.H. Le, A note on Terai’s conjecture concerning Pythagorean numbers, Proc. Japan Acad. Ser. A, 74 (5), 80–81, 1998.
- [3] H.W. Gould, Tables of combinatorial identities, https://web.archive.org/web/ 20190629193344/http://www.math.wvu.edu/~gould/ (Gould’s personal webpage).
- [4] J.Y. Hu and H. Zhang, A conjecture concerning primitive Pythagorean triples, Int. J. Appl. Math. Stat. 52 (7), 38–42, 2014.
- [5] M.H. Le, A note on the Diophantine equation $x^2+b^y=c^z$, Acta Arith. 71 (3), 253–257, 1995.
- [6] M.H. Le, On Terai’s conjecture concerning Pythagorean numbers, Acta Arith. 100 (1), 41–45, 2001.
- [7] M.H. Le and G. Soydan, A brief survey on the generazlized Lebesgue-Ramanujan- Nagell equation, Surv. Math. Appl. 15, 473–523, 2020.
- [8] R. Lidl and H. Neiderreiter, Finite Fields, Cambridge Univ. Press, Cambridge, 1996.
- [9] L.J. Mordell, Diophantine equations, Academic Press, London, 1969.
- [10] G. Soydan, M. Demirci, I.N. Cangül and A. Togbé, On the conjecture of Jeśmanowicz, Int. J. Appl. Math. Stat. 56 (6), 46–72, 2017.
- [11] N. Terai, A note on the Diophantine equation $x^2+q^m=p^n$, Acta Arith. 63 (4), 351–358, 1993.
- [12] P.Z. Yuan, The Diophantine equation $x^2+b^y=c^z$, J. Sichuan Univ. Nat. Sci. 41 525–530, 1998 (in Chinese).
- [13] P.Z. Yuan and J.B. Wang, On the Diophantine equation $x^2+b^y=c^z$, Acta Arith. 84 (2), 145–147, 1998.
Abstract
References
- [1] M.A. Bennett, J.S. Ellenberg and N.C. Ng, The Diophantine equation $A^4+2^\delta B^2=C^n$, Int. J. Number Theory, 6 (2), 311–338, 2010.
- [2] X.G. Chen and M.H. Le, A note on Terai’s conjecture concerning Pythagorean numbers, Proc. Japan Acad. Ser. A, 74 (5), 80–81, 1998.
- [3] H.W. Gould, Tables of combinatorial identities, https://web.archive.org/web/ 20190629193344/http://www.math.wvu.edu/~gould/ (Gould’s personal webpage).
- [4] J.Y. Hu and H. Zhang, A conjecture concerning primitive Pythagorean triples, Int. J. Appl. Math. Stat. 52 (7), 38–42, 2014.
- [5] M.H. Le, A note on the Diophantine equation $x^2+b^y=c^z$, Acta Arith. 71 (3), 253–257, 1995.
- [6] M.H. Le, On Terai’s conjecture concerning Pythagorean numbers, Acta Arith. 100 (1), 41–45, 2001.
- [7] M.H. Le and G. Soydan, A brief survey on the generazlized Lebesgue-Ramanujan- Nagell equation, Surv. Math. Appl. 15, 473–523, 2020.
- [8] R. Lidl and H. Neiderreiter, Finite Fields, Cambridge Univ. Press, Cambridge, 1996.
- [9] L.J. Mordell, Diophantine equations, Academic Press, London, 1969.
- [10] G. Soydan, M. Demirci, I.N. Cangül and A. Togbé, On the conjecture of Jeśmanowicz, Int. J. Appl. Math. Stat. 56 (6), 46–72, 2017.
- [11] N. Terai, A note on the Diophantine equation $x^2+q^m=p^n$, Acta Arith. 63 (4), 351–358, 1993.
- [12] P.Z. Yuan, The Diophantine equation $x^2+b^y=c^z$, J. Sichuan Univ. Nat. Sci. 41 525–530, 1998 (in Chinese).
- [13] P.Z. Yuan and J.B. Wang, On the Diophantine equation $x^2+b^y=c^z$, Acta Arith. 84 (2), 145–147, 1998.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Mathematics |
| Authors | |
| Publication Date | August 6, 2021 |
| Published in Issue | Year 2021 |