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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