NONE 20260504120748419 wseas/wseas content-registration-form+ja@crossref.org content-registration-form WSEAS TRANSACTIONS ON MATHEMATICS 1109-2769 2224-2880 Vipawadee Moonchaisook Department of Mathematics, Faculty of Science and Technology Surindra Rajabhat University Surin 32000 THAILAND This paper investigates the structural behavior of the exponential Pell-type Diophantine equation $$x^2 = 11y^2 - 7^t$$ for $$t \in \mathbb{Z}^+$$ and provides a complete characterization of its solvability. Using modular arguments and properties of the Legendre symbol, we show that the equation admits no integer solutions when the exponent $$t$$ is even, whereas infinitely many nontrivial solutions arise when $$t$$ is odd. In the solvable case, Brahmagupta’s composition law and the fundamental unit of the Pell equation $$x^2 - 11y^2 = 1$$ are employed to construct explicit solution sequences. These solutions satisfy second-order linear recurrences, admit closed-form expressions, and preserve quadratic and quartic algebraic invariants. Numerical computations support the theoretical results and highlight the sharp contrast between the solvable and unsolvable cases. This work extends classical Pell-type theory to exponential Diophantine equations with mixed quadratic–exponential structure. 05 04 2026 05 04 2026 57 7 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/23206.2026.25.7 https://wseas.com/journals/mathematics/2026/a145106-2314.pdf M. J. Jacobson, Jr. and H. C. Williams, Solving the Pell Equation, CMS Books in Mathematics, Springer Science+Business Media, New York, 2009. ISBN: 978-0-387-84922-5. https://doi.org/10.1007/978-0-387-84923-2. E. J. Barbeau, Pell’s Equation, Problem Books in Mathematics, Springer–Verlag, New York, 2003. ISBN: 0-387-95529-1. D. A. Marcus, Number Fields, Springer–Verlag, New York, 1977. ISBN: 978-0-387-90279-1. W. 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