What is 100,000 as a sum of some number of distinct squares?

Question

I cannot find the solution to this problem. It is part of a larger homework question but I can't go on until I solve this question.

Answer 1

HINT: Find the largest square less than $100,000$, subtract it, and see what remains. (Note that in general you can’t expect this procedure necessarily to work, but it’s an obvious first thing to try, and in this case it’s helpful.)

Answer 2

Let's start by placing the largest number we can as a summand, (because this will make the problem "smaller" for us, although it may not be the correct aproach, but it won't hurt to try).

Taking the square root we find it is $316^2=99,856$. So now we need to write $100,000-99,856=144$ as a square or sum of squares, which is easy as $12^2=144$.

So $100,000=12^2+316^2$

Answer 3

A brute-force approach by computer yields $$100000=12^2+316^2=100^2+300^2=180^2+260^2$$

Answer 4

The complete answer, to express $10^5$ as sum of distinct squares in as many ways as possible, has to do with the factorization of $10$ in the Gaussian numbers, $10=(3+i)(3-i)=i(1-i)^2(2+i)(2-i)$. I’m too lazy and groggy to work it out completely, but since $(3+i)^2=8+6i$ and $(3+i)^4=28+96i$, you can replace the $10^4$ in the equation $10^5=100^2+300^2$ by $80^2+60^2$ as well as by $28^2+96^2$.