Show that the smallest quadratic non residue of an odd prime p is itself prime. Hint: Assume to the contrary and use Legendre Symbols

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- Nov 12th 2009, 10:28 PMscubasteve123quadratic non residue
Show that the smallest quadratic non residue of an odd prime p is itself prime. Hint: Assume to the contrary and use Legendre Symbols

- Nov 13th 2009, 09:36 AMchiph588@
Assume not so let $\displaystyle n $ be the smallest QNR such that $\displaystyle n=ab $. Then $\displaystyle -1=\left(\frac{n}{p}\right)=\left(\frac{ab}{p}\righ t) = \left(\frac{a}{p}\right)\left(\frac{b}{p}\right) = (1)(1) = 1 $ since $\displaystyle a,b < n $. Hence a contradiction.