Is the product of adjacent primes of the form $36x^2-1$?

Question

If $p$ and $q$ are primes such that $p-q=2$, will $pq=36x^2-1$ be always true for some natural number $x$?

Answer 1

Every natural no, 5 onwards exists in 1 of the following 6 forms: $6x-1,6x,6x+1,6x+2,6x+3,6x+4$ for some natural no. x

Of these, $6x-1$ and $6x+1$ are the only ones that can possibly be prime. As both the primes can not be of the same form (difference would be a multiple of 6),

$p=6x-1$ and $q=6x+1$

$pq=36x^2-1$

This rule works only from 5 onwards, so there is an exception, which is (3,5) whose product, 15 can not be written as $36x^2-1$