On the Diophantine Equation $\left(9d^2 + 1\right)^x + \left(16d^2 - 1\right)^y = (5d)^z$ Regarding Terai's Conjecture

Year 2024, , 72 - 84, 30.06.2024

Tuba Çokoksen , Murat Alan

https://doi.org/10.53570/jnt.1479551

Abstract

This study proves that the Diophantine equation $\left(9d^2+1\right)^x+\left(16d^2-1\right)^y=(5d)^z$ has a unique positive integer solution $(x,y,z)=(1,1,2)$, for all $d>1$. The proof employs elementary number theory techniques, including linear forms in two logarithms and Zsigmondy's Primitive Divisor Theorem, specifically when $d$ is not divisible by $5$. In cases where $d$ is divisible by $5$, an alternative method utilizing linear forms in p-adic logarithms is applied.

Keywords

Terai, Diophantine equations, primitive divisor theorem

References

• T. N. Shorey, R. Tijdeman, Exponential diophantine equations, Cambridge University Press, Cambridge, 1986.
• W. Sierpinski, On the equation $3^x +4^y =5^z$, Wiadomości Matematyczne 1 (2) (1956) 194-195.
• L. Jesmanowicz, Several remarks on pythagorean numbers, Wiadomości Matematyczne 1 (2) (1955) 196-202.
• N. Terai, The Diophantine equation $a^x+b^y=c^z$ and $abc \neq 0$, Proceedings of the Japan Academy Series A Mathematical Sciences 70 (1994) 22-26.
• N. Terai, T. Hibino, On the exponential Diophantine equation, International Journal of Algebra 6 (23) (2012) 1135-1146.
• T. Miyazaki, N. Terai, On the exponential Diophantine equation, Bulletin of the Australian Mathematical Society 90 (1) (2014) 9-19.
• N. Terai, T. Hibino, On the exponential Diophantine equation $(12m^2+ 1)^x+(13m^2- 1)^y=(5m)^z$, International Journal of Algebra 9 (6) (2015) 261-272.
• R. Fu, H. Yang, On the exponential Diophantine equation, Periodica Mathematica Hungarica, 75 (2) (2017) 143-149.
• X. Pan, A note on the exponential Diophantine equation $(am^2+1)^x+(bm^2-1)^y =(cm)^z$ in colloquium mathematicum, Instytut Matematyczny Polskiej Akademii Nauk 149 (2017) 265-273.
• M. Alan, On the exponential Diophantine equation $(18m^2+1)^x+(7m^2−1)^y= (5m)^z$, Turkish Journal of Mathematics 42 (4) (2018) 1990-1999.
• E. Kizildere, T. Miyazaki, G. Soydan, On the Diophantine equation $((c +1)m^2+ 1)^x+(cm^2-1)^y= (am)^z$, Turkish Journal of Mathematics 42 (5) (2018) 2690-2698.
• N-J. Deng, D-Y. Wu, P-Z. Yuan, The exponential Diophantine equation $(3am^2-1)^x+(a(a-3)m^2+1)^y=(am)^z$, Turkish Journal of Mathematics 43 (5) (2019) 2561-2567.
• N. Terai, On the exponential Diophantine equation, Annales Mathematicae et Informaticae 52 (2020) 243-253.
• E. Kızıldere, G. Soydan, On the exponential Diophantine equation $(5pn^2−1)^x+p(p−5)n^2+1)^y=(pn)^z$, Honam Mathematical Journal 42 (2020) 139-150.
• N. Terai, Y. Shinsho, On the exponential Diophantine equation $(3m^2 +1)^x +(qm^2-1)^y = (rm)^z$, SUT Journal of Mathematics 56 (2020) 147-158.
• N. Terai, Y. Shinsho, On the exponential Diophantine equation $(4m^2 +1)^x +(45m^2-1)^y = (7m)^z$, International Journal of Algebra 15 (4) (2021) 233-241.
• M. Alan, R-G. Biratlı, On the exponential Diophantine equation $(6m^2 +1)^x+(3m^2 −1)^y = (3m)^z$, Fundamental Journal of Mathematics and Applications 5 (3) (2022) 174-180.
• J. Y. He, J. G. Luo, S. L. Fei, On the exponential Diophantine equation $(a(a-l)m^2+1)^x+(alm^2−1)^y=(am)^z$, Aims Mathematics 7 (4) (2022) 7187-7198.
• E. Hasanalizade, A note on the exponential Diophantine equation $(44m² + 1)^x+ (5m² - 1)^ y= (7m)^z$, Integers: Electronic Journal of Combinatorial Number Theory 23 (1) (2023) 10 pages.
• M. Laurent , Linear forms in two logarithms and interpolation determinants II, Acta Arithmetica 133 (2008) 325-348.
• Y. Bugeaud, Linear forms in p-adic logarithms and the Diophantine equation $(x^n-1)/(x-1) = y^q$, Mathematical Proceedings of the Cambridge Philosophical Society 127 (1999) 373-381.
• K. Zsigmondy, Zur theorie der potenzreste, Monatshefte für Mathematik 3 (1892) 265-284.
• M. Alan, On the exponential Diophantine equation $(18m^2 + 1)^x + (7m^2 − 1)^y = (5m)^z$, Turkish Journal of Mathematics 42 (4) (2018) 1990-1999.
• A-Y. Khinchin, Continued fractions, University of Chicago Press, Chicago, 1964.