Problem 672

Erdős Problem #672 [status: verifiable; formalized: yes]. Can the product of an arithmetic progression of positive integers $n,n+d,\ldots,n+(k-1)d$ of length $k\geq 4$ (with $(n,d)=1$) be a perfect power? Current best: Considering the question of whether the product of an arithmetic progression of length $k$ can be equal to an $\ell$th power: {UL} {LI}Euler proved this is impossible when $k=4$ and $\ell=2$,{/LI} {LI}Obl\'{a}th [Ob51] proved this is impossible when $(k,l)=(5,2),(3,3),(3,4),(3,5)$.{/LI} {LI}Marszalek [Ma85] proved that this is only possible for $k\ll_d 1$, where $d$ is the common difference of the arithmetic progression.{/LI} {LI}Gy\"{o}ry, Hajdu, and Saradha [GHS04] proved this is impossible for $4\leq k\leq 5$. {LI}Bennett, Bruin, Gy\"{o}ry, and Hajdu [BBGH06] proved this is impossible for $4\leq k\leq 11$, and also impossible for $k$ sufficiently large depending only on the number of prime divisors of $d$.{/LI} {LI} Gy\"{o}ry, Hajdu, and Pint\'{e}r [GHP09] have proved this is impossible for $4\leq k\leq 34$.{/LI} {/UL} Jakob F\"{u}hrer has observed this is possible for integers in general, for example $(-6)\cdot(-1)\cdot 4\cdot 9=6^3$. Prize: no. Tags: number theory.