Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 1G / 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 givesfor positive odd coprime $a,n$; the sign exponents add modulo two, proving Jacobi reciprocity from Legendre reciprocity. Since $959=7\cdot 137$, $(5/7)=-1$ and $(5/137)=(2/5)=-1$, so $(5/959)=1$. Nevertheless $5$ is a nonresidue modulo both prime factors, so $x^2\equiv 5(\mathrm{mod}959)$ has no solution.