Let Q be a set of prime numbers, not necessarily finite. For a positive integer n consider its prime factorization: define p(n) to be the sum of all the exponents and q(n) to be the sum of 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>0 independent of the set Q such that for any positive integer N>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(63)=3, q(63)=3, p(2022)=3, q(2022)=1.) number theoryprime factorizationfunctionprime numbersprimes