Find all numbers $p$ such that all six numbers $p$, $p+2$, $p+6$, $p+8$, $p+12$, $p+14$ are primes
I know that $p=5$ works, but I don't know how to find all values for $p$, if any?
Find all numbers $p$ such that all six numbers $p$, $p+2$, $p+6$, $p+8$, $p+12$, $p+14$ are primes
I know that $p=5$ works, but I don't know how to find all values for $p$, if any?
Hint:
Consider these numbers modulo $5$ $p,p+2,p+1,p+3,p+2,p+4$ This is a complete set of residues. Hence, one of these number must be divisible by $5$.
This means that $p\leq 5$, otherwise all these numbers would be greater than $5$ and one of them will be a multiple of $5$. Hence, one of them is composite. (contradicting the fact that all of them are primes.)