Abstract: 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.

Keywords: Pell equation, exponential Diophantine equation, Brahmagupta’s composition, solvability criterion, recurrence relation, algebraic invariants

Pages: 57-66