Contributions towards a conjecture of Erdős on perfect powers in arithmetic progression

Abstract

Let $n,d,k\geq2,b,y$ and $\ell\geq3$ be positive integers with the greatest prime factor of b not exceeding k. It is proved that the equation $n (n+d) \dotsb (n+(k-1)d)=b y^{\ell}$ has no solution if d exceeds d₁, where d₁ equals 30 if $\ell =3$; 950 if $\ell =4$; $5\times 10^4$ if $\ell=5$ or 6; 10⁸ if $\ell=7$, 8, 9 or 10; 10¹⁵ if $\ell \geq 11$. This confirms a conjecture of Erdős on the above equation for a large number of values of d.