In mathematics, a Størmer number or arc-cotangent irreducible number is a positive integer for which the greatest prime factor of is greater than or equal to . They are named after
Carl Størmer.
More precisely, the
natural density of the Størmer numbers lies between 0.5324 and 0.905.
It has been conjectured that their natural density is the
natural logarithm of 2, approximately 0.693, but this remains unproven.[2]
Because the Størmer numbers have positive density, the Størmer numbers form a
large set.
Application
The Størmer numbers arise in connection with the problem of representing the
Gregory numbers (
arctangents of
rational numbers) as sums of Gregory numbers for integers (arctangents of
unit fractions). The Gregory number may be decomposed by repeatedly multiplying the
Gaussian integer by numbers of the form , in order to cancel prime factors from the imaginary part; here is chosen to be a Størmer number such that is divisible by .[3]
^Everest, Graham; Harman, Glyn (2008), "On primitive divisors of ", Number theory and polynomials, London Math. Soc. Lecture Note Ser., vol. 352, Cambridge Univ. Press, Cambridge, pp. 142–154,
arXiv:math/0701234,
doi:
10.1017/CBO9780511721274.011,
MR2428520. See in particular Theorem 1.4 and Conjecture 1.5.
^Conway, John H.;
Guy, R. K. (1996), The Book of Numbers, New York: Copernicus Press, pp. 245–248. See in particular p. 245, para. 3.