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

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