Exercise 3.2.14 (Property of twin primes)

Proof. Since $q=p+2$, one of the twin primes $p,q$ is of the form $4n+1$. Then the law of quadratic reciprocity shows that

If there is an integer $a$ such that $p\mid a^2-q$, the congruence $x^2\equiv q(\mathit{mod}p)$ has a solution $a$, therefore $\left(\frac{q}{p}\right)=1$, hence $\left(\frac{p}{q}\right)=1$. This shows that there is an integer $b$ such that $q\mid b^2-p$.

Similarly, if there is an integer $b$ such that $q\mid b^2-p$, then $\left(\frac{p}{q}\right)=1$, hence $\left(\frac{q}{p}\right)=1$, and there is an integer $a$ such that $p\mid a^2-q$ □