Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 1I / Solution

If the positive odd integer $n$ has prime factorizationits Jacobi symbol iswhere each factor is a Legendre symbol; by convention $(a/1)=1$. Thus the symbol is zero exactly when $\gcd (a,n)>1$.

The quadratic reciprocity law for the Jacobi symbol says that for coprime positive odd integers $m,n$,To prove it, write $m={\prod }_i{p}_i^{e_i}$ and $n={\prod }_j{q}_j^{f_j}$. The quadratic reciprocity law for distinct odd primes statesMultiplying this identity over $i,j$ with multiplicities $e_i{f}_j$ gives the required product of Jacobi symbols. Its sign exponent modulo two isbecause multiplication of odd integers adds the quantities $(r-1)/2$ modulo two. This proves the law.

For the requested value, two applications of reciprocity givesince $2019\equiv 7(\mathrm{mod}503)$, $503\equiv 6(\mathrm{mod}7)$, and $6$ is a quadratic nonresidue modulo $7$.