I know a proof for a similar problem, but I've hit a wall.
Let me show you the groundwork . . . maybe you can finish it.
The product of four positive integers in arithmetic progression
cannot be the square of an integer.
Let the four numbers be: .
Suppose their product is a square.
Then we have: .
And we have: .
And this is supposed to lead to a contradiction . . . but I can't find it.