  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 $x\geq 3$. The proof relies on properties
of random subsets of the prime numbers.
