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For some fixed $n$ define the quadratic form $$Q(x,y) = x^2 + n y^2.$$

I think that if $Q$ represents $m$ in two different ways then $m$ is composite.

I can prove this for $n$ prime. I was hoping someone could give me a hint towards proving this result for general $n$? Also would be interested in generalizations if any are known! Thanks a lot.

2 Answers 2

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Below is Lucas' classic proof, from his Theorie des nombres, 1891, as described in section 215 of Mathews: Theory of Numbers. enter image description here enter image description here

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John Brillhart published a paper about this in the American Mathematical Monthly some time in the past year.

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    I think it's [A Note on Euler's Factoring Problem](http://www.jstor.org/pss/40391253).2011-04-15
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    Yes, that's the paper I had in mind.2011-04-15
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    I cannot access this but thank you for the comment.2011-04-15
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    @quanta, that's what libraries are for. Another approach would be to send email to Professor Brillhart. But maybe Bill's answer has given you what you need.2011-04-16
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    @quanta: Brillhart's paper also employs Lucas' proof.2011-04-16