Determine the set of primes for which 11 is a quadratic residue.

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- Jul 21st 2009, 11:55 PMCairoQuadratic Residues
Determine the set of primes for which 11 is a quadratic residue.

- Jul 22nd 2009, 12:35 PMTheAbstractionist
First, note that 11 is a quadratic residue mod 2 (e.g. $\displaystyle 1^2\equiv11\,(\bmod\,2)).$

Let $\displaystyle p$ and $\displaystyle q$ be odd primes. There is a result (equivalent to the law of quadratic reciprocity) that says that $\displaystyle (11\,|\,p)=(11\,|\,q)$ if and only if $\displaystyle p\equiv\pm q\,(\bmod\,44).$ So you only have to determine all odd primes $\displaystyle p < 44$ for which $\displaystyle (11\,|\,p)=1$ All other odd primes for which 11 is a quadratic residue will congruent to plus or miinus one of these. - Jul 22nd 2009, 11:43 PMCairoThanks
Thanks.

Do you know where I can find more information on the result that you quoted? I've never seen this before. - Jul 23rd 2009, 11:25 AMTheAbstractionist
I came across it in H.E. Rose,

*A Course in Number Theory*, p.67:

Quote:

Theorem 2.2*Let $\displaystyle p$ and $\displaystyle q$ be distinct odd primes and $\displaystyle a\ge1.$ The quadratic reciprocity law is equivalent to*

*if $\displaystyle p\equiv\pm q\,(\bmod\,4a)$ then $\displaystyle (a/p)=(a/q).$*