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, etc. (sequence A005528 in OEIS)

The Størmer numbers arise in connection with the problem of representing Gregory numbers ta / b as sums of Gregory numbers for integers: "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 number p (n is the smallest number for which n² + 1 is divisible by p)."[1]

References

1. ^ Conway & Guy (1996): 245, ¶ 3

* 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.