I wonder that, I made these observations from my previous study on product of consecutive integers. I am looking the solutions of these kind of equations.
$(1)$ Is $x(x+1)(x+2)...(x+[\text{any-odd-integer}]) = y^2$ has solutions or not?. If exits, how to list them?
$(2)$ Is $x(x+d)(x+2d) = y^2$ has infinitely many solutions or not? If there, how to find them.
$(3)$ For $k \ne 2,4$, can we have solutions of $x(x+1)(x+2)...(x+k-1)+Q = t^2$, where Q is a rational number.