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I'm interested in nontrivial solutions of Diophantine equations of the type $$a^2b^3 = \frac{n!}{(n-k)!} $$$$a^2b^3 = \frac{c!}{(c-k)!} $$

For various values of k fixed, and of course $a,b,n \in \mathbb{Z^+}$$a,b,c \in \mathbb{Z^+}$

Does anyone have any insight into this type of equation or a good reference for further reading? My search is being swamped by irrelevant results.

Edit: I changed n to c to emphasize that I am looking for a,b,c that solve this equation. Thus for k= 1, the equation becomes $a^2b^3 = c$, which clearly has infinity many solutions.

I'm interested in nontrivial solutions of Diophantine equations of the type $$a^2b^3 = \frac{n!}{(n-k)!} $$

For various values of k fixed, and of course $a,b,n \in \mathbb{Z^+}$

Does anyone have any insight into this type of equation or a good reference for further reading? My search is being swamped by irrelevant results.

I'm interested in nontrivial solutions of Diophantine equations of the type $$a^2b^3 = \frac{c!}{(c-k)!} $$

For various values of k fixed, and of course $a,b,c \in \mathbb{Z^+}$

Does anyone have any insight into this type of equation or a good reference for further reading? My search is being swamped by irrelevant results.

Edit: I changed n to c to emphasize that I am looking for a,b,c that solve this equation. Thus for k= 1, the equation becomes $a^2b^3 = c$, which clearly has infinity many solutions.

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G G
G G
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Diophantine equation of a factorial type

I'm interested in nontrivial solutions of Diophantine equations of the type $$a^2b^3 = \frac{n!}{(n-k)!} $$

For various values of k fixed, and of course $a,b,n \in \mathbb{Z^+}$

Does anyone have any insight into this type of equation or a good reference for further reading? My search is being swamped by irrelevant results.