Schinzel circle

Illustration of a Schinzel circle

Schinzel circles are a set of circles with a given number of integer points on the circumference of the circle.

If the number n of points on the circumference of the circle is even, n = 2k, then a Schinzel circle is given by:

[math]\displaystyle{ \left(x-\frac{1}{2}\right)^2 + y^2 = \frac{1}{4} 5^{k-1} }[/math],

whereas for odd n = 2k + 1,

[math]\displaystyle{ \left(x-\frac{1}{3}\right)^2 + y^2 = \frac{1}{9} 5^{2k} }[/math].