Størmer number

In mathematics, a Størmer number or arc-cotangent irreducible number, named after Carl Størmer, is a positive integer n for which the greatest prime factor of (n² + 1) meets or exceeds 2n.

The first few Størmer numbers are:

J. Todd proved that this sequence is neither finite nor cofinite.

The Størmer numbers arise in connection with the problem of representing the Gregory numbers (arctangents of rational numbers) $G_{a/b}=\arctan\frac{b}{a}$ as sums of Gregory numbers for integers (arctangents of unit fractions). The Gregory number $G_{a/b}$ may be decomposed by repeatedly multiplying the Gaussian integer $a+bi$ by numbers of the form $n\pm i$ , in order to cancel prime factors p from the imaginary part; here $n$ is chosen to be a Størmer number such that $n^2+1$ is divisible by $p$ .