How many squares does it take to express
every whole number as the sum of squares?
We saw that two was not enough in
Sums of Two Squares. Perhaps three? Or four?
Well, three is not enough, but almost. The only
whole numbers which cannot be written as the
sum of 3 squares are numbers of the form
4^{m}(8k+7). So you will have problems
writing 7, 15, or 28 as the sum of three squares.
But every whole number can be written as the
sum of four squares!
Accordingly,
7=2^{2}+1^{2}+1^{2}+1^{2},
and
15=3^{2}+2^{2}+1^{2}+1^{2}.
Presentation Suggestions:
Have the class pick their favorite number and
write it as the sum of four squares.
The Math Behind the Fact:
The sum of 4 squares result was stated by
Gerard, Fermat, and Diophantus(?), but first proved by
Lagrange in 1770. It is a classic result in number theory.
How to Cite this Page:
Su, Francis E., et al. "Sums of Three and Four Squares."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
