Størmer number

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 "n"2 + 1 meets or exceeds 2"n". The first few Størmer numbers are 1, 2, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 19, 20, etc. OEIS|id=A005528

The Størmer numbers arise in connection with the problem of representing Gregory numbers "t""a" / "b" as sums of Gregory numbers for integers: "To find Størmer's decomposition for "t""a" / "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 number "p" ("n" is the smallest number for which "n"2 + 1 is divisible by "p")." [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.


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