I'm sure the argument goes like this. Pick a large integral value of R. The number of (x,y) points within the circle of radius R centered at the origin is approximately pi R^2 (and tends to pi R^2 as R tends to infinity). There are about R^2 distinct values between 0 and R^2 (the maximum of x^2+y^2 within that circle). Thus, the average number of points for each such value is pi R^2 / R^2 which is just pi. On Wed, Jul 18, 2018 at 12:52 PM Colin Wright <math_fun@solipsys.co.uk> wrote:

I recently saw the following claim:

"On the average, the number of ways of expressing a positive integer n as a sum of two integral squares, x^2 + y^2 = n, is pi"

Can anyone confirm or deny this?

Thanks.

Colin -- Some of you may have had occasion to run into mathematicians and to wonder therefore how they got that way, ... -- Tom Lehrer

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Without loss of generality x >= y, therefore: x^2 - y^2 = n - 2y^2 can be written in the same number of ways, pi on average. The left hand side is a factorization of n - 2y^2: (x+y)(x-y) = n - 2y^2 So there are pi (perhaps trivial) factorizations of n - 2y^2, on average, for all the varying x and y pairs (i.e. not fixed y). Really? On 7/18/18 12:51 , Colin Wright wrote:

I recently saw the following claim:

"On the average, the number of ways of expressing a positive integer n as a sum of two integral squares, x^2 + y^2 = n, is pi"

Can anyone confirm or deny this?

Thanks.

Colin