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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Related http://math.stackexchange.com/questions/122048/1-is-a-quadratic-residue-modulo-p-if-and-only-if-p-equiv-1-pmod4 – lab bhattacharjee May 19 '13 at 06:31
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1You're asking for $-1$ to be a quadratic residue. This is discussed in any text that does quadratic residues. – Gerry Myerson May 19 '13 at 06:32
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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 $\,x^2=-1\pmod m\,$.
DonAntonio
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