Problem 679

Erdős Problem #679: declared status 'open'. Formalized: no. Let $\epsilon>0$ and $\omega(n)$ count the number of distinct prime factors of $n$. Are there infinitely many values of $n$ such that$$\omega(n-k) < (1+\epsilon)\frac{\log k}{\log\log k}$$for all $k<n$ which are sufficiently large depending on $\epsilon$ only? Can one show the stronger version with$$\omega(n-k) < \frac{\log k}{\log\log k}+O(1)$$is false? Prize: no. Tags: number theory.