Question 91
Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorisation; define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of all the exponents corresponding only to primes in $Q$. A positive integer $n$ is called *special* if $p(n) + p(n + 1)$ and $q(n) + q(n + 1)$ are both even integers. Prove that there is a constant $c \gt 0$ independent of the set $Q$ such that for any positive integer $N \gt 100$, the number of special integers in $[1, N]$ is at least $cN$.
(For example, if $Q = \{3, 7\}$, then $p(42) = 3$, $q(42) = 2$, $p(315) = 4$, $q(315) = 3$)
(For example, if $Q = \{3, 7\}$, then $p(42) = 3$, $q(42) = 2$, $p(315) = 4$, $q(315) = 3$)