In the Fun Fact Sums of Two Squares, we’ve seen which numbers can be written as the sum of two squares. For instance, 11 cannot, but 13 can (as 32+22). A related question, with a surprising answer, is: on average, how many ways can a number can be written as the sum of two squares?
We should clarify what we mean by average. Let W(N) is the number of ways to write N as the sum of two squares. Thus W(11)=0, and W(13)=8 (as sums of squares of all possible combinations of +/-3 and +/-2 , in either order).
So if A(N) is the average of the numbers W(1), W(2), …, W(N), then A(N) is the average number of ways the first N numbers can be written as the sum of two squares. Then it makes sense to take the limitof A(N) as N goes to infinity to get the “average” number of ways to write a number as the sum of two squares, over all positive whole numbers.
A surprising fact is that this limit exists, and it is Pi!
This might be presented after a discussion of lattice points in Pick’s Theorem.
The Math Behind the Fact:
The proof is as neat as the result! Every solution (x,y) to x2+y2=N can be thought of as a lattice point in the plane, i.e., a point with integer coordinates. Such a lattice point lies on a circle of radius Sqrt(N).
Therefore, the sum of W(1) through W(N) counts the number of lattice points in the plane inside or on a circle of radius Sqrt(N) (except for the origin), and the average A(N) is this number of lattice points divided by N. But as N goes to infinity, the number of lattice points inside this circle is approximately the area of the circle, hence Pi times the radius squared: Pi*Sqrt(N)2 or Pi*N. Therefore A(N) is approximately Pi, and this approximation gets better and better as N goes to infinity!
How to Cite this Page:
Su, Francis E., et al. “Sums of Two Squares Ways.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
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