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 3^{2}+2^{2}). 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 limit
of 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!
Presentation Suggestions:
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 x^{2}+y^{2}=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!
Counting the the number of lattice points inside a circle is known as Gauss' circle problem.
Also, see several other Fun Facts about sums of two squares or
formulas for pi.
How to Cite this Page:
Su, Francis E., et al. "Sums of Two Squares Ways."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
