Puzzle time - sums of two squares

[Horace]

Missed this one the first time. So you're looking for the number of pairs of integers lying on the circle of radius sqrt(n) since a^2 + b^2 is distance from origin squared. The problem then becomes the number of times that circle passes through a point composed of two integers. By symmetry and restriction to positive integers you would only look at one eighth of the circle. Average number of times of an unbounded n implies this number does not increase without limit, and may be cyclical. I suspect the problem can be solved brute force by observing this cycle or this limit, but there is probably an elegant way to consider how many times this circle passes through integer vertexes.

Actually I suppose the problem is asking for a function of n rather than a specific number. So it then becomes the number of times a circular arc of length (pi * 2 * sqrt(n))/8 passes through integer points on the plane.

[jon-nyc]

Really we're looking for the expected value across all the integers from 1 to a zillion.

[Klaus]

Hm, that number looks pretty familiar. A favorite of Larry's in fact.

[jon-nyc]

Yes, by way of that hint, this one collapses into the Gauss Circle Problem.

[Doctor Phibes]

Klaus and Jon doing math.

The sums of two squares.

[jon-nyc]

It’s kind of a cool result. It’s framed as a kind of number theory problem and the answer is a transcendental number associated with geometry.

[Horace]

It is a cool result. But I'm not sure there's any low-math, non-brute-force way to get there. The connection between sum of squares and a circle is not very far towards the answer.

[jon-nyc]

Unless you recognize it as the Gauss Circle Problem, which has a well-known answer.

[jon-nyc]

And I would add an intuitive answer.

[Horace]

Yeah I guess the area thing would be intuitive and from there the circumference thing is also intuitive.