Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 22, No. 2, pp. 143-148 (2006)

On $D$ so that $x^2 - Dy^2 = \pm m$

John P. Robertson

Actuarial and Economic Services Division, National Counsil on Compensation Insurance, Baca Raton

Abstract: We prove that for any integer m (different from 0, +2, -2), there are infinitely many positive integers D for which the form x^2 - Dy^2 primitively represents m, -m, and -1. We do this by constructing an infinite sequence of such D's associated with each m.

Also, when m is odd, we relate the existence of additional such D's to well-known conjectures.

Keywords: Generalized Pell equation, simultaneous Pell equations, representation

Classification (MSC2000): 11D09; 11D85

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