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I have a book that proves the three laws of quadratic reciprocity for primes i.e $\left(\frac{-1}p\right)=1$ if $p=1\mod4$ and $-1$ if $p=3\mod4$, where $p$ is prime, etc.
But in the end the book says that the three laws are also true for arbitary odd integers.

Why is that true?

Thank you.

Shaun
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TheGeometer
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  • There's an extension of the Legendre symbol (the one you know about, where $p$ is prime) to something called the Jacobi symbol. You can do a search for that, and probably find all your questions answered. Then you can post an answer to your own question. That's encouraged on this site. – Gerry Myerson Jun 10 '14 at 11:24

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