Elementary Methods in Number Theory Exercise 1.4.12

Prove that if $p$ and $q$ are twin primes greater than 3,then $p+q$ is divisible by 12.

Proof:According to Elementary Methods in Number Theory Exercise 1.4.6,if $p$ and $p+2$ are twin prime numbers,and both of them are greater than 3,then $p+4+3k(k\in\mathbf{2N})$ can't be a prime number.

 

So $p+2+3k,p+4+3k(k\in\mathbf{2N})$ are not twin prime numbers,and $p+4+3k,p+6+3k(k\in\mathbf{2N})$ are not twin prime numbers.So it is only possible that $p+6+3k,p+8+3k(k\in\mathbf{2N})$ are twin prime numbers.We know that

\begin{equation}\label{eq:7632}
[(p+6+3k)+(p+8+3k)]=(12+6k)+(p+p+2)
\end{equation}
And $12|(12+6k)$.We know that $12|5+7$.According to \ref{eq:7632},and mathematical induction,$12|(p+q)(p,q> 3)$