On the Diophantine Equation $(8r^2+1)^x+(r^2-1)^y=(3r)^z$ Regarding Terai's Conjecture

Year 2024, Volume: 7 Issue: 4, 199 - 211, 31.12.2024

Tuba Çokoksen , Murat Alan

https://doi.org/10.33434/cams.1561789

Abstract

This study establishes that the sole positive integer solution to the exponential Diophantine equation $(8r^2+1)^x+(r^2-1)^y=(3r)^z$ is $(x,y,z)=(1,1,2)$ for all $r>1$. The proof employs elementary techniques from number theory, a classification method, and Zsigmondy's Primitive Divisor Theorem.

Keywords

Diophantine equations, Primitive divisor theorem, Terai’s conjecture

References

• W. Sierpinski, On the equation $3^x +4^y =5^z$, Wiad. Mat., 1 (1956), 194–195.
• L. Jesmanowicz, Several remarks on Pythagorean numbers, Wiad. Mat., 1(2) (1955), 196–202.
• N. Terai, The Diophantine equation $a^x+b^y=c^z$, Proc. Japan Acad. Ser. A Math. Sci., 70 (1994), 22-26.
• N. Terai, T. Hibino, \emph{On the exponential Diophantine equation}, Int. J. Algebra, 6(23) (2012), 1135–1146.
• T. Miyazaki, N. Terai, On the exponential Diophantine equation, Bull. Aust. Math. Soc., 90(1) (2014), 9–19.
• N. Terai, T. Hibino, On the exponential Diophantine equation $(12m^2+ 1)^x+(13m^2- 1)^y=(5m)^z$, Int. J. Algebra, 9(6) (2015), 261–272.
• R. Fu, H. Yang, On the exponential Diophantine equation, Period. Math. Hungar., 75(2) (2017), 143–149.
• X. Pan, A note on the exponential Diophantine equation, Colloq. Math., 149 (2017), 265–273.
• M. Alan, On the exponential Diophantine equation $(18m^2+1)^x+(7m^2−1)^y= (5m)^z$, Turkish J. Math., 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 J. Math., 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 J. Math., 43(5) (2019), 2561 – 2567.
• N. Terai, On the exponential Diophantine equation, Ann. Math. Inform., 52 (2020), 243–253.
• E. Kızıldere, G. Soydan, On the Diophantine equation $(5pn^2−1)^x+(p(p−5)n^2+1)^y=(pn)^z$, Honam Math. J., 42 (2020), 139–150.
• N. Terai, Y. Shinsho, On the exponential Diophantine equation $(3m^2 +1)^x +(qm^2-1)^y = (rm)^z$, SUT J. Math., 56 (2020) 147-158.
• N. Terai, Y. Shinsho, On the exponential Diophantine equation $(4m^2 +1)^x +(45m^2-1)^y = (7m)^z$, Int. J. Algebra, 15(4) (2021), 233-241.
• M. Alan, R.G. Biratlı, On the exponential Diophantine equation $(6m^2 +1)^x+(3m^2 −1)^y = (3m)^z$, Fundam. J. Math. Appl., 5(3) (2022), 174-180.
• S. Fei, J. Luo, A Note on the Exponential Diophantine Equation $(rlm^2-1)^x+(r (r-l) m^2+ 1)^y=(rm)^z$, Bull. Braz. Math. Soc. (N.S.), 53 (2022), 1499-1517.
• E. Hasanalizade, A note on the exponential Diophantine equation $(44m + 1)^x+ (5m - 1)^ y= (7m)^z$, Integers, 23 (2023), 1.
• T. Çokoksen, M. Alan, On the Diophantine equation $(9d^2 + 1)^x + (16d^2 − 1)^y = (5d)^z$ Regarding Terai's Conjecture, J. New Theory, 47 (2024), 72-84.
• A. Çağman, Repdigits as sums of three Half-companion Pell numbers}, Miskolc Math. Notes, 24(2) (2023), 687-697.
• A. Çağman, K. Polat, On a Diophantine equation related to the difference of two Pell numbers, Contrib. Math., 3 (2021), 37-42.
• A. Çağman, Explicit Solutions of Powers of Three as Sums of Three Pell Numbers Based on Baker’s Type Inequalities, TJI, 5(1) (2021), 93-103.
• M. Le, Some exponential Diophantine equations. I. The equation $d_1x^2- d_2y^2=\lambda k^z$, J. Number Theory, 55 (1995), 209-221.
• Y. Bugeaud, T. Shorey, On the number of solutions of the generalized Ramanujan-Nagell equation, J. Reine Angew. Math., 539 (2001), 55-74.
• Y. Bilu, G. Hanrot, P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math., 539 (2001), 75-122.
• P. M. Voutier, Primitive divisors of Lucas and Lehmer sequences, Math. Comp., 64 (1995), 869-888.
• K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., 3 (1892), 265–284.
• L. K. Hua, Introduction to Number Theory, Science Publishing Co, (1957).
• J. H. E. Cohn, Square Fibonacci numbers, J. Lond. Math. Soc. (2), (1964), 109-113.