= Solution
For $n=\prod p_i^{e_i}$ odd, define $(a/n)=\prod(a/p_i)^{e_i}$. Multiplying <quadratic reciprocity> for all prime-factor pairs gives
$$
\left(\frac an\right)\left(\frac na\right)=(-1)^{((a-1)/2)((n-1)/2)}
$$
for positive odd coprime $a,n$; the sign exponents add modulo two, proving Jacobi reciprocity from Legendre reciprocity. Since $959=7\cdot137$, $(5/7)=-1$ and $(5/137)=(2/5)=-1$, so $(5/959)=1$. Nevertheless $5$ is a nonresidue modulo both prime factors, so $x^2\equiv5\pmod{959}$ has no solution.
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