I want to find the Legendre symbol : $\left(\frac{3}{2^{2^n}+1}\right)$ for any positive n.
Also I want to find the Legendre symbol for: $\left(\frac{5}{2^{2^n}+1}\right)$, when $n>1$
Solution:
Using Gaussian reciprocity law: $\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}\frac{q-1}{2}}$, we get $\left(\frac{3}{2^{2^n}+1}\right)$=$\left(\frac{2^{2^n}+1}{3}\right)$.Similarly, $\left(\frac{5}{2^{2^n}+1}\right)$=$\left(\frac{2^{2^n}+1}{5}\right)$.
But I am unable to proceed any further. Thanks in advance.