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

[摘要] Let $n,d,kgeq2,b,y$ and $ellgeq3$ 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 $ell =4$; $5imes 10^4$ 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.

[学科分类] 数学（综合）