Størmer number


A Størmer number or arc-cotangent irreducible number is a positive integer n for which the greatest prime factor of n2+1 exceeds 2n. The first few Størmer numbers are 2, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 39, 40, 42, 44, 45, 48, 49, etc., listed in A005528 of Sloane’s OEIS. Weakening the inequalityMathworldPlanetmath from gpf(n2+1)>2n to gpf(n2+1)2n makes no difference other than admitting 1 to the list (and possibly changing index offsets accordingly).

The Størmer numbers arise in connection with the problem of representing Gregory numbers tab as sums of Gregory numbers for integers. Conway and Guy explain in their book thus: “To find Størmer’s decomposition for ta/b, you repeatedly multiply a+bi by numbers n±i for which n is a Størmer number and the sign is chosen so that you can cancel the corresponding prime numberMathworldPlanetmath p (n is the smallest number for which n2+1 is divisible by p).”

Størmer numbers are named after the Norwegian physicist Carl Størmer (http://planetmath.org/CarlStormer).

References

  • 1 John H. Conway & R. K. Guy, The Book of Numbers. New York: Copernicus Press (1996): 245 - 248.
  • 2 J. Todd, “A problem on arc tangent relations”, Amer. Math. Monthly, 56 (1949): 517 - 528.
Title Størmer number
Canonical name StormerNumber
Date of creation 2013-03-22 17:52:16
Last modified on 2013-03-22 17:52:16
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 5
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A51
Synonym Stormer number
Synonym Störmer number
Synonym arc-cotangent irreducible number