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Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 25, No. 2, pp. 155-164 (2009)
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On D so that x^2 - Dy^2 represents m and -m and not -1

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John P. Robertson

National Council on Compensation Insurance, USA

**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

**Full text of the article:**

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© 2009
FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition
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