Consecutive Sophie Germain Primes cannot be Pair of Twin Primes

Theorem

Let $p$ and $p + 2$ be twin primes.

Then unless $p = 3$ it is not possible for both $p$ and $p + 2$ to be Sophie Germain primes.

Proof

First it is noted that $3$ and $5$ twin primes which are both Sophie Germain.

Prime numbers greater than $3$ are of the form $6 n - 1$ and $6 n + 1$.

Thus a pair of twin primes is of the form $\left({6 n - 1, 6 n + 1}\right)$.

The result follows from Sophie Germain Prime cannot be 6n+1.

$\blacksquare$