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W(n) < w(n+1) < w(n+2) for infinitely many n

Source: 4th German TST 2006, written on 1 April 2006, problem 1

April 3, 2006
Eulernumber theory proposednumber theory

Problem Statement

For any positive integer nn, let w(n)w\left(n\right) denote the number of different prime divisors of the number nn. (For instance, w(12)=2w\left(12\right)=2.) Show that there exist infinitely many positive integers nn such that w(n)<w(n+1)<w(n+2)w\left(n\right)<w\left(n+1\right)<w\left(n+2\right).
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