A power saving below two-thirds for equal radicals of consecutive products

Let F(X) count positive integer pairs m < n ≤ X with rad(m(m+1)) = rad(n(n+1)). We prove F(X) ≪_ε X^(109/165+ε) for every ε > 0, improving the earlier exponent 2/3. A four-corner parametrization divides these pairs into thirteen classes. Uniform estimates for those classes combine with a finite exponent calculation and one choice of error terms.