Q:

Prove that 2 is a nonresidue of any integer = ± 3 (mod 8), without using Gauss’s lemma.

Prove that 2 is a nonresidue of any integer = ± 3 (mod 8), without using Gauss’s lemma. 
Help me!
And what is the relation between fourier transform and Mobius Function?

A:

Gauss proved this in his Disquisitiones Arithmeticae. Suppose that $x^2\equiv 2\pmod n$ with $n\equiv\pm3\pmod 8$. We can assume $x$ is odd and $3\le x\le n-2$.
Then $x^2-2=mn$ where $0<m<n$ and $m\equiv\pm3\pmod 8$. Now, use infinite descent.

