Abstract. We consider the Diophantine equation of the form x2 – Dy2 = c, where c | 2D, gcd(x,y) = 1, and provide criteria for solutions in terms of congruence conditions on the fundamental solution of the Pell Equation x2 – Dy2 = 1. The proofs are elementary, using only basic properties of simple continued fractions. The results generalize various criteria for such solutions, and expose the central norm, defined by the infrastructure of the underlying real quadratic field, as the foundational key that binds all the elements.
Keywords: Quadratic Diophantine equations, continued fractions, central norms, fundamental unit.
AMS Subject classification: Primary: 11D09, 11R11, 11A55; Secondary: 11R29.
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