
Legendre symbol
Hey guys, I need a bit of help here.
1. Show that the equation $\displaystyle x^271y^2=19$ has no integer solution using Legendre symbol.
So $\displaystyle \left(\frac{a}{p}\right) \equiv a^{(p1)/2}\ \pmod{p}$
$\displaystyle \left(\frac{71}{19}\right) \equiv 71^{(191)/2}\ \pmod{19} $
$\displaystyle 71^9\ \pmod{19} = 18 = 1$
So because it equals $\displaystyle 1$, $\displaystyle 71$ is a quadratic nonresidue modulo $\displaystyle 19$ and so there is no integer solution to $\displaystyle x^271y^2=19$.
Is this enough to show no integer solutions?
2. Evaluate the Legendre symbol $\displaystyle \left(\frac{p^21}{p}\right)$ for odd prime $\displaystyle p$.
So $\displaystyle \left(\frac{p^21}{p}\right) \equiv (p^21)^{(p1)/2}\ \pmod{p}$
What is the formal way to go from here?
I have evaluated this for several cases of odd prime p and kind of see a pattern.
Also, this value alternates between 1 and 1 for odd numbers.
Any help guys?
Thanks for your help.

Re: Legendre symbol
1. $\displaystyle x^271y^2=19 \implies x^2\equiv 19 \pmod{71}$
$\displaystyle \left(\frac{19}{71}\right)=$$\displaystyle \left(\frac{71}{19}\right)=1$
So, 19 is a quadratic nonresidue modulo 71 and therefore the equation has no integer solutions.
2. $\displaystyle \left(\frac{p^21}{p}\right)=\left(\frac{(p+1)(p1)}{p}\right)$
$\displaystyle =\left(\frac{1}{p}\right)\left(\frac{1}{p}\right)$
$\displaystyle =(1)^{(p1)/2}$

Re: Legendre symbol
Oh cool, thanks alex, I understand it!