Comments on: Calculating whether A is a quadratic residue mod N, N composite? http://www.andersoncompositesquadron.org/composite-properties/calculating-whether-a-is-a-quadratic-residue-mod-n-n-composite Sun, 11 Jul 2010 12:55:59 -0400 http://wordpress.org/?v=2.8.4 hourly 1 By: cheeser1234567890 http://www.andersoncompositesquadron.org/composite-properties/calculating-whether-a-is-a-quadratic-residue-mod-n-n-composite/comment-page-1#comment-213 cheeser1234567890 Tue, 29 Sep 2009 09:14:59 +0000 http://www.andersoncompositesquadron.org/composite-properties/calculating-whether-a-is-a-quadratic-residue-mod-n-n-composite#comment-213 I think you're confused: http://en.wikipedia.org/wiki/Legendre_symbol http://en.wikipedia.org/wiki/Euler%27s_criterion http://en.wikipedia.org/wiki/Jacobi_symbol The Jacobi symbol is what allows the Legendre symbol to be used in a way that makes sense for composite numbers. It's the way to do it. When I say the way, I really mean the way, not just a way. As far as I've ever seen, there aren't better ways to do it, unless the Jacobi symbol itself simplifies into something more sensible. What you're hoping is that Euler's criterion holds for a composite modulus. This is not generally true, and when it is true, I believe what you're looking for is Euler-Jacobi pseudoprimes. See: http://en.wikipedia.org/wiki/Euler-Jacobi_pseudoprime<br><b>References : </b><br> I think you’re confused:

http://en.wikipedia.org/wiki/Legendre_symbol
http://en.wikipedia.org/wiki/Euler%27s_criterion
http://en.wikipedia.org/wiki/Jacobi_symbol

The Jacobi symbol is what allows the Legendre symbol to be used in a way that makes sense for composite numbers. It’s the way to do it. When I say the way, I really mean the way, not just a way. As far as I’ve ever seen, there aren’t better ways to do it, unless the Jacobi symbol itself simplifies into something more sensible.

What you’re hoping is that Euler’s criterion holds for a composite modulus. This is not generally true, and when it is true, I believe what you’re looking for is Euler-Jacobi pseudoprimes. See:
http://en.wikipedia.org/wiki/Euler-Jacobi_pseudoprime
References :

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