The eighteenth-century mathematician Joseph Louis Lagrange proved that every positive integer is the sum of, at most, four squares. Find at least one way to write as the sum of three squares.

Source: NCTM Mathematics Teacher 2008

**SOLUTION
**

a sum of four squares

We want a sum of three squares, so we try , etc.

not a perfect square

Continuing in this slow and tedious fashion we will find more solutions

**Answer**:

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Also from Lagrange:

A number n is the sum of two squares if and only if the primefactors >2 of n that are == 3 mod(4) have even multiplicity. So after you have substracted the first square (44² etc.) and get 72=2*2*2*3*3 you know 72 is the sum of two squares. (3 appear and even number of times)

If you had gotten 62=3*31 instead etc, you know it was not possible, since 31 is =3 (mod4) and have odd multiplicity.