Research Article

On the Diophantine Equation $(a^n-1)(b^n-1)(c^n-1)=x^2$

Number: 55 June 30, 2026

On the Diophantine Equation $(a^n-1)(b^n-1)(c^n-1)=x^2$

Abstract

In this paper, we study the exponential Diophantine equation $(a^n-1)(b^n-1)(c^n-1)=x^2$ in nonnegative integers for certain fixed values of $a$, $b$, and $c$ with $ 1$<$a$<$b$<$c$. Our aim is to extend the classical two-factor framework $(a^n-1)(b^n-1)=x^2$ to the corresponding three-factor setting. We first establish a general nonexistence criterion based on $2$-adic valuations and the lifting-the-exponent lemma. As an application, we prove that the equation $(2^n-1)(5^n-1)(7^n-1)=x^2$ has no positive integer solutions. Moreover, we derive a more general result covering a family of triples $(a,b,c)$ under suitable parity conditions. Furthermore, we prove that the equation $(2^n-1)(3^n-1)(5^n-1)=x^2$ has a unique solution $(n, x)=(2, 24)$ under a certain congruence restriction on $n$.

Keywords

References

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Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Publication Date

June 30, 2026

Submission Date

March 18, 2026

Acceptance Date

June 11, 2026

Published in Issue

Year 2026 Number: 55

APA
Alan, M., & Yildirim, Y. (2026). On the Diophantine Equation $(a^n-1)(b^n-1)(c^n-1)=x^2$. Journal of New Theory, 55, 38-46. https://doi.org/10.53570/jnt.1912478
AMA
1.Alan M, Yildirim Y. On the Diophantine Equation $(a^n-1)(b^n-1)(c^n-1)=x^2$. JNT. 2026;(55):38-46. doi:10.53570/jnt.1912478
Chicago
Alan, Murat, and Yunus Yildirim. 2026. “On the Diophantine Equation $(a^n-1)(b^n-1)(c^n-1)=x^2$”. Journal of New Theory, nos. 55: 38-46. https://doi.org/10.53570/jnt.1912478.
EndNote
Alan M, Yildirim Y (June 1, 2026) On the Diophantine Equation $(a^n-1)(b^n-1)(c^n-1)=x^2$. Journal of New Theory 55 38–46.
IEEE
[1]M. Alan and Y. Yildirim, “On the Diophantine Equation $(a^n-1)(b^n-1)(c^n-1)=x^2$”, JNT, no. 55, pp. 38–46, June 2026, doi: 10.53570/jnt.1912478.
ISNAD
Alan, Murat - Yildirim, Yunus. “On the Diophantine Equation $(a^n-1)(b^n-1)(c^n-1)=x^2$”. Journal of New Theory. 55 (June 1, 2026): 38-46. https://doi.org/10.53570/jnt.1912478.
JAMA
1.Alan M, Yildirim Y. On the Diophantine Equation $(a^n-1)(b^n-1)(c^n-1)=x^2$. JNT. 2026;:38–46.
MLA
Alan, Murat, and Yunus Yildirim. “On the Diophantine Equation $(a^n-1)(b^n-1)(c^n-1)=x^2$”. Journal of New Theory, no. 55, June 2026, pp. 38-46, doi:10.53570/jnt.1912478.
Vancouver
1.Murat Alan, Yunus Yildirim. On the Diophantine Equation $(a^n-1)(b^n-1)(c^n-1)=x^2$. JNT. 2026 Jun. 1;(55):38-46. doi:10.53570/jnt.1912478

 

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