In a recent Wrong But Useful podcast, @reflectivemaths (who is Dave Gale in real life) asked the audience to:
Prove that the sum of five consecutive square numbers is never a square.
This one's a bit easier than it looks: I chose to call the middle number of the five $n$. That makes my sum:
$(n-2)^2 + (n-1)^2 + n^2 + (n+1)^2 + (n+2)^2$
The reason I've done it like that is that I get a lot of cancelling if I do! In fact, when I expand the brackets, all of the $n$ terms vanish and I'm left with $5n^2 + 10 = 5(n^2 + 2)$.
That can only be a square if $n^2 + 2$ is a multiple of five - so $n^2$ must end in 3 or 8. Sadly, all squares end in either 0, 1, 4, 5, 6, or 9, so there's no such $n$.