Am I me or my evil twin?

Since $p$ and $q$ are twin primes, and $q > p$ , $q = p+2$ . Considering $p \equiv 1 \pmod 3$ would imply that $q \equiv 0 \pmod 3$ or $3 | p$ which is a contradiction since $q$ is prime. Considering $p \equiv 2 \pmod 3$ would imply that $q \equiv 1 \pmod 3$ and thus is the only option for $p$ and $q$ to be twin primes.

Since $p \equiv 2 \pmod 3$ and $p \equiv 1 \pmod 2$ we can easily deduce from the chinese remainder theorem that $p = 6n + 5$ , since $q = p+2$ , $q = 6n + 7$

n=1 is a solution