On D so that x^2 - Dy^2 represents m and -m and not -1

Abstract: For $m = 25$, $100$, $p$, $2p$, $4p$, or $2 p^2$, where $p$ is prime, we show that there is at most one positive nonsquare integer $D$ so that the form $x^2 - D y^2$ primitively represents $m$ and $-m$ and does not represent $-1$. We give support for a conjecture that for any $m > 1$ not listed above, there are infinitely many $D$ so that the form $x^2 - Dy^2$ primitively represents $m$ and $-m$ and does not represent $-1$.

Keywords: Generalized Pell equation, simultaneous Pell equations, representation

Classification (MSC2000): 11D09; 11D85

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