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While investigating something about square numbers, I started to wonder which numbers $n$ have quadratic residues of $n-1$. Obviously all $n$ of the form $m^2+1$ will work. But there are other numbers such as $13$, is there a pattern to them? Any insights on this would be great.

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In very short:

The equation $\,x^2=-1\pmod p\,,\;p\,$ a prime, has solution iff $\,p=1\pmod 4\,$ or $\,p=2\,$ (this is already a very nice exercise), and from here, with the prime factorization of $\,n\in\Bbb N\,$ , you get the corresponding condition for $\,x^2=-1\pmod m\,$.

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  • $\begingroup$ What's $p=2$? Chopped liver? $\endgroup$ – Gerry Myerson May 19 '13 at 6:46
  • $\begingroup$ Oops...that's what happens to even primes. Corrected and thanks. $\endgroup$ – DonAntonio May 19 '13 at 6:51

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