We say that a positive integer is good if it has an even number of prime factors. Does there exist an arbitrarily large sequence of consecutive good numbers?
In fact, this problem came from another one: Show that there exists an infinite set $S$ of positive integers such that the sum of any two distinct elements of S has an even number of distinct prime factors.
If the first statement about long sequences of good numbers is true, then I can finish the second problem as follows: Choose $s_{1}\in \mathbb{N}$ randomly. Then, we will define the increasing sequence $s_{n}$ inductively. Take a sequence of $s_{n-1}+1$ consecutive good numbers. If $a$ is the smallest one in this sequence, take $s_{n}=a$. Then set $S=\{s_{n}\}^{\infty}_{n=1}$.