18 Jul
2018
18 Jul
'18
4:32 p.m.
On 18/07/2018 20:51, 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"
The average number of ways to express numbers from 1 to N as sums of two squares = 1/N times the total number of ways to express numbers from 1 to N as sums of two squares = 1/N times the total number of (x,y) for which 0 < x^2+y^2 <= N = 1/N times (the total number of (x,y) inside the circle X^2 + Y^2 = N, minus 1) = 1/N times (the area inside the circle X^2 + Y^2 = N, + O(N)) = 1/N times (the area inside the circle X^2 + Y^2 = sqrt(N)^2, + O(N)) = 1/N ( pi sqrt(N)^2 + O(N)) = pi + O(1/N). -- g