Schinzel circle

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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].

[1] [2]

References

  1. Weisstein, Eric. "Schinzel Circle". Wolfram MathWorld. http://mathworld.wolfram.com/SchinzelCircle.html. Retrieved 19 August 2015. 
  2. Mutanguha, Jean Pierre. "Schinzel's Theorem". http://euler.genepeer.com/schinzels-theorem/.