Schinzel's theorem

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:

$\left(x-{\frac {1}{2}}\right)^{2}+y^{2}={\frac {1}{4}}5^{k-1}$,

whereas for odd n = 2k + 1,

$\left(x-{\frac {1}{3}}\right)^{2}+y^{2}={\frac {1}{9}}5^{2k}$.