Why is the sum of five consecutive squares never a square?

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 $n$.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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