Finding primes $p$ such that $ \dfrac {p+1}2$ and $\dfrac{p-1}4$ are primes How many odd primes $p$ are there such that both $ \dfrac {p+1}2$ and $\dfrac{p-1}4$ are primes ?

One of $p,p+1,p+2$ is divisible by $3$, dividing by 2 or 4 doesn't change this so for all of these numbers to be prime we must have either $p=3$, $\frac{p+1}{2}=3$ or $\frac{p-1}{4}=3$, solve for $p$ and you'll find that $p=13$ is the only prime with the property you wanted 
