# Størmer number

A Størmer number or arc-cotangent irreducible number is a positive integer $n$ for which the greatest prime factor of $n^{2}+1$ 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 inequality from $\textrm{gpf}(n^{2}+1)>2n$ to $\textrm{gpf}(n^{2}+1)\geq 2n$ 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 $t_{\frac{a}{b}}$ as sums of Gregory numbers for integers. Conway and Guy explain in their book thus: “To find Størmer’s decomposition for $t_{a/b}$, you repeatedly multiply $a+bi$ by numbers $n\pm i$ 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 $p$).”

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 StormerNumber 2013-03-22 17:52:16 2013-03-22 17:52:16 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Definition msc 11A51 Stormer number Störmer number arc-cotangent irreducible number