Large Gaps between Sums of Two Squareful Numbers

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.