The Tijdeman–Zagier conjecture for products

Let $x,y,n,m,d,s$ be positive integers, with $x$ and $y$ coprime. A product of degree $d$ and spread $s$ is a product of $d$ positive integers whose maximum minus minimum is $s$.

Tijdeman–Zagier conjecture for products. The integer $x^n\pm y^m$ is not a product of degree $d$ satisfying

$$2<d\leq\min(n,m)$$

and of spread $s$ satisfying

$$\frac{1}{n}+\frac{1}{m}+\frac{1+s}{d}<1.$$

This generalizes the classical Tijdeman–Zagier, or Beal, conjecture by permitting the third term to be a product rather than a single power. The paper reports computational testing but no proof or disproof, so the conjecture remains open.