Minimal Pythagorean circles
Let us define a Pythagorean circle, as a circle in the plane, centered at origin, with one or more points of the circle having both x and y coordinates as integers. Given the symmetry of the situation, the number of such points is divisible by four, so let us divide the number of points by four and call this the degree of the Pythagorean circle. If we have a Pythagorean circle of degree X, where there are no Pythagorean circles with a smaller radius and degree X, we call that a minimal Pythagorean circle for that degree. The radius of such a circle might not be integer, but its radius squared will always be (to find the radius squared, take one of the points with integer x and y, and calculate x^2+y^2, and this will be the radius squared, by simple trigonometry.)
Let us list a few minimal Pythagorean circles, and their radius squared:
| R^2 | Degree | Points in the first quadrant |
|---|---|---|
| 1 | 1 | (0,1) |
| 5 | 2 | (1,2) (2,1) |
| 25 | 3 | (0,5) (3,4) (4,3) |
| 65 | 4 | (1,8) (4,7) (7,4) (8,1) |
| 625 | 5 | (0,25) (7,24) (15,20) (20,15) (24,7) |
| 325 | 6 | (1,18) (6,17) (10,15) (15,10) (17,6) (18,1) |
Now looking at R^2 you might notice a pattern. Lets see the factorizations of the radius squared of these minimal Pythagorean circles:
| Degree | R^2 factorized |
|---|---|
| 1 | 1 |
| 2 | 5 |
| 3 | 5^2 |
| 4 | 5*13 |
| 5 | 5^4 |
| 6 | 5^2*13 |
| 7 | 5^6 |
| 8 | 5*13*17 |
| 9 | 5^2*13^2 |
| 10 | 5^4*13 |
| 12 | 5^2*13*17 |
| 14 | 5^6*13 |
Now, the challenges: What is the next prime factor to show itself in this kind of factorization? What is the minimal Pythagorean circle for degree 11 and 13? What is the complete pattern, and is there a formula for finding the minimal Pythagorean circle for a given degree?
Update: added this diagram of a pythagorean circle of degree 3, with radius squared 25 (and radius 5): 
Update2: a hint is that no sum of two square numbers can be written as 4x+3, where x is integer