# Thread: Proof By Induction: Infinitely many pairs of consecutive pprime-ish numbers

1. ## Proof By Induction: Infinitely many pairs of consecutive pprime-ish numbers

Call an integer pprime-ish if each of its prime factors occurs with power two or higher. Prove by induction that there are in finitely many pairs of consecutive pprime-ish positive integers.

I know how to prove that there are infinitely number primes, but am unsure of how to prove this.

2. ## Re: Proof By Induction: Infinitely many pairs of consecutive pprime-ish numbers

Hint: If k and k+1 are pprime-ish, then so are 4(k)(k+1) and 4k(k+1)+1.