On the Diophantine Equation (an − 1)(bn − 1)(cn − 1) = x2
Journal of New Theory, no.55, pp.38-46, 2026 (TRDizin)
- Publication Type: Article / Article
- Volume: Issue: 55
- Publication Date: 2026
- Doi Number: 10.53570/jnt.1912478
- Journal Name: Journal of New Theory
- Journal Indexes: TR DİZİN (ULAKBİM)
- Page Numbers: pp.38-46
- Open Archive Collection: AVESIS Open Access Collection
- Yıldız Technical University Affiliated: Yes
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 1 < a < b < c. 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.