Here's an easy result, two sums of five distinct squares that give a square:
42 + 62 + 102 + 142 + 862 = 882
862 + 1292 + 2152 + 3012 + 8812 = 9682
They are examples I'm using, pulled from the same series that generates an infinity of them.
Web searching I DID find three squares summed to give a square as there is a known identity, but haven't found a square the sum of five distinct squares or more. So suspect these techniques are new.
Innovation is a critical sign. You can't fake it.
Finding new things in number theory is very exciting for people who love numbers. But it's also incredibly hard, so I like to search thoroughly, and get as much help as possible before trumpeting too much.
My web searching might miss something! So very curious to see if anyone can find prior research showing how to sum an arbitrary number of distinct squares to get a square.
Or you can just try to generate such a series yourself: test your math ability.
If you can't easily find your own sum of 5 distinct squares that is itself a square, why not?
Maybe more fun to try that before looking at how I do it.
So yeah, I posted about it, after first posting the easy algebra for getting an arbitrary number of squares that give a square.
The solution is surprisingly easy and can generate an infinity of such solutions or of any arbitrary number of squares. You could sum a hundred squares or even a thousand to get a square if you wished, though I'd think it better to let a computer do it!
It's SO easy to do though that I don't think of it as fascinating for difficulty, but only for whether it was known or not.
It's something you can check.
Do you love numbers? I do. So I'm curious about them.
Looking for help from other curious people who love numbers.