Problem #672: Can the product of an arithmetic progression of positive...

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Progress

Erdős believed not. Erdős and Selfridge [ErSe75] proved that the product of consecutive integers is never a perfect power.

The theory of Pell equations implies that there are infinitely many pairs $n,d$ with $(n,d)=1$ such that $n(n+d)(n+2d)$ is a square.

Considering the question of whether the product of an arithmetic progression of length $k$ can be equal to an $\ell$th power:

• Euler proved this is impossible when $k=4$ and $\ell=2$,


• Obláth [Ob51] proved this is impossible when $(k,l)=(5,2),(3,3),(3,4),(3,5)$.


• Marszalek [Ma85] proved that this is only possible for $k\ll_d 1$, where $d$ is the common difference of the arithmetic progression.


• Györy, Hajdu, and Saradha [GHS04] proved this is impossible for $4\leq k\leq 5$.
  
• Bennett, Bruin, Györy, 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 $d$.


• Györy, Hajdu, and Pintér [GHP09] have proved this is impossible for $4\leq k\leq 34$.

Jakob Führer has observed this is possible for integers in general, for example $(-6)\cdot(-1)\cdot 4\cdot 9=6^3$.