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:

1, 2, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 19, 20, ... (sequence A005528 in OEIS).

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 $p$.^[1]

Notes

1. Conway & Guy (1996): 245, ¶ 3

References

• John H. Conway & R. K. Guy, The Book of Numbers. New York: Copernicus Press (1996): 245–248.
• J. Todd, "A problem on arc tangent relations", Amer. Math. Monthly, 56 (1949): 517–528.