How many twin primes are there whose sum is a power of a prime?

Question

How many twin primes are there whose sum is a power of a prime? I have started with the form p+(2+p), which reduces to 2(p+1), which suggests the number must be even. Where could I go from here?

Answer 1

If $(p,p+2)$ are a pair of twin primes, their sum is even, so if it equals to the power of a prime then that prime must be $2$. (Can you see why?) Furthermore, if $p>3$ then modulo $3$ we have $p\equiv -1$ and $p+2\equiv 1$, since in any other case $3$ would divide either $p$ or $p+2$, which is absurd. This means that their sum $p+(p+2)\equiv(-1)+1\equiv 0$ modulo $3$. But $p+(p+2)$ is supposed to be a power of $2$, so how can it possibly be divisible by $3$?

Thus, the only answer is $(3,5)$.

Answer 2

As @Catalin has noted. Any twin primes bigger than 3 and 5 are congruent to $\pm1$ modulo 6. Thus their sum is divisible by 6 an not a prime power. So 3 and 5 are the only twin primes adding up to a prime power.