Past exam of the options II course of the University of Oxford / 2004 / C / C2 / b / Solution

Quadratic reciprocity with the prime three givesThe latter symbol is $+1$ when $p\equiv 1(\mathrm{mod}3)$ and $-1$ when $p\equiv 2(\mathrm{mod}3)$. A prime greater than three occupies one of the classes $1,5,7,11$ modulo twelve. Combining its residues modulo three and four gives signs $+1,-1,-1,+1$ respectively. Hence