Large gaps between sums of two squareful numbers

Abstract

Let $M(x)$ be the length of the largest subinterval of $[1,x]$ which does not contain any sums of two squareful numbers. We prove a lower bound \[ M(x)\gg \frac{\ln x}{(\ln\ln x)^2} \] for all $x\ge 3$. The proof relies on properties of random subsets of the prime numbers.

8 pages, misprints and mistakes corrected in version 2