Square Root of $5$ mod $10^{9}+7$

Question

$My$ $Current$ $Knowledge:$ We can find it if 5 is a $Quadratic$ $residue$ modulo p and where p is prime and we can check it using $Euler$ $criterion$. I cannot able to find the root(5)mod 1000000007. Since 10^9+7 is prime we will able to find its solution.

Answer 1

There is no solution because the Jacobi/Legendre symbol is $(5 | 10^9+7)=-1$. With $p=10^9+7$ you compute $5^{\frac{p-1}{2}}\equiv -1 \pmod p.$

Using the law of quadratic reciprocity, $5 \equiv 1 \pmod 4,\;$ and the other properties the Jacobi/Legendre symbol $$\left(\frac{5}{p}\right)=\left(\frac{p}{5}\right)=\left(\frac{p\bmod 5}{5}\right)= \left(\frac{2}{5}\right)=-1 $$