What question did this study set out to answer?

The aim is to explore the Diophantine equation $(a^n-1)(b^n-1)(c^n-1)=x^2$ for nonnegative integers.

July 2, 2026Open Access

On the Diophantine Equation (aⁿ-1) (bⁿ-1) (cⁿ-1) =x²

Key Points

The aim is to explore the Diophantine equation $(a^n-1)(b^n-1)(c^n-1)=x^2$ for nonnegative integers.
Established a nonexistence criterion using 2-adic valuations and the lifting-the-exponent lemma.
Proved specific cases such as $(2^n-1)(5^n-1)(7^n-1)=x^2$ has no positive integer solutions.
Derived a unique solution for $(2^n-1)(3^n-1)(5^n-1)=x^2$ under congruence conditions.
No positive integer solutions exist for $(2^n-1)(5^n-1)(7^n-1)=x^2$.
The unique solution $(n, x)=(2, 24)$ was found for $(2^n-1)(3^n-1)(5^n-1)=x^2$.

Abstract

In this paper, we study the exponential Diophantine equation (aⁿ-1) (bⁿ-1) (cⁿ-1) =x² in nonnegative integers for certain fixed values of a, b, and c with 1abc. Our aim is to extend the classical two-factor framework (aⁿ-1) (bⁿ-1) =x² 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 (2ⁿ-1) (5ⁿ-1) (7ⁿ-1) =x² 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 (2ⁿ-1) (3ⁿ-1) (5ⁿ-1) =x² has a unique solution (n, x) = (2, 24) under a certain congruence restriction on n.

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