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

Tuba Çokoksen; Murat Alan

doi:10.53570/jnt.1479551

EN

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

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

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Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Authors

Tuba Çokoksen ^*
0009-0004-3164-1211
Türkiye

Murat Alan
0000-0003-2031-2725
Türkiye

Publication Date

June 30, 2024

Submission Date

May 6, 2024

Acceptance Date

June 26, 2024

Published in Issue

Year 2024 Number: 47

DOI

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

IZ

https://izlik.org/JA48DC99JH

APA

Çokoksen, T., & Alan, M. (2024). On the Diophantine Equation $\left(9d^2 + 1\right)^x + \left(16d^2 - 1\right)^y = (5d)^z$ Regarding Terai’s Conjecture. Journal of New Theory, 47, 72-84. https://doi.org/10.53570/jnt.1479551

AMA

1.Çokoksen T, Alan M. On the Diophantine Equation $\left(9d^2 + 1\right)^x + \left(16d^2 - 1\right)^y = (5d)^z$ Regarding Terai’s Conjecture. JNT. 2024;(47):72-84. doi:10.53570/jnt.1479551

Chicago

Çokoksen, Tuba, and Murat Alan. 2024. “On the Diophantine Equation $\left(9d^2 + 1\right)^x + \left(16d^2 - 1\right)^y = (5d)^z$ Regarding Terai’s Conjecture”. Journal of New Theory, nos. 47: 72-84. https://doi.org/10.53570/jnt.1479551.

EndNote

Çokoksen T, Alan M (June 1, 2024) On the Diophantine Equation $\left(9d^2 + 1\right)^x + \left(16d^2 - 1\right)^y = (5d)^z$ Regarding Terai’s Conjecture. Journal of New Theory 47 72–84.

IEEE

[1]T. Çokoksen and M. Alan, “On the Diophantine Equation $\left(9d^2 + 1\right)^x + \left(16d^2 - 1\right)^y = (5d)^z$ Regarding Terai’s Conjecture”, JNT, no. 47, pp. 72–84, June 2024, doi: 10.53570/jnt.1479551.

ISNAD

Çokoksen, Tuba - Alan, Murat. “On the Diophantine Equation $\left(9d^2 + 1\right)^x + \left(16d^2 - 1\right)^y = (5d)^z$ Regarding Terai’s Conjecture”. Journal of New Theory. 47 (June 1, 2024): 72-84. https://doi.org/10.53570/jnt.1479551.

JAMA

1.Çokoksen T, Alan M. On the Diophantine Equation $\left(9d^2 + 1\right)^x + \left(16d^2 - 1\right)^y = (5d)^z$ Regarding Terai’s Conjecture. JNT. 2024;:72–84.

MLA

Çokoksen, Tuba, and Murat Alan. “On the Diophantine Equation $\left(9d^2 + 1\right)^x + \left(16d^2 - 1\right)^y = (5d)^z$ Regarding Terai’s Conjecture”. Journal of New Theory, no. 47, June 2024, pp. 72-84, doi:10.53570/jnt.1479551.

Vancouver

1.Tuba Çokoksen, Murat Alan. On the Diophantine Equation $\left(9d^2 + 1\right)^x + \left(16d^2 - 1\right)^y = (5d)^z$ Regarding Terai’s Conjecture. JNT. 2024 Jun. 1;(47):72-84. doi:10.53570/jnt.1479551

Cited By

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

Communications in Advanced Mathematical Sciences

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