Puzzle time - sums of two squares
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On average, how many ways are there to write a positive number n as the sum of two squares?
In other words, suppose n is a random integer between 1 and a zillion. What is the expected number of ordered pairs (i,j) of integers such that n = i^2+j ^2 ? For example, if n = 25 we get not only 25 = 0^2 + 5^2 and 25 = 3^2 + 4^2, but also 25 = 5^2 + 0^2 = 3^2 + (-4)2 etc., twelve ways in all.
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HINT:
Let f(n) be the number of ways to write n as the sum of two squares. Then the average of f(n) for n = 1,2,...,Z is T/n where T is the total number of integer pairs (i,j) for which i^2 + j^2 is at most Z. What do these pairs (i,j) look like as Cartesian coordinates on the plane?
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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.
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Klaus and Jon doing math.
The sums of two squares.
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Unless you recognize it as the Gauss Circle Problem, which has a well-known answer.
