Størmer number

In mathematics, a Størmer number or arc-cotangent irreducible number is a positive integer $n$ for which the greatest prime factor of $n^{2}+1$ is greater than or equal to $2n$ . They are named after Carl Størmer.

Sequence

The first few Størmer numbers are:

1, 2, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 19, 20, ... (sequence A005528 in the OEIS).

Density

John Todd proved that this sequence is neither finite nor cofinite.[1]

Unsolved problem in mathematics:

What is the natural density of the Størmer numbers?

(more unsolved problems in mathematics)

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) $G_{a/b}=\arctan {\frac {b}{a}}$ as sums of Gregory numbers for integers (arctangents of unit fractions). The Gregory number $G_{a/b}$ may be decomposed by repeatedly multiplying the Gaussian integer $a+bi$ by numbers of the form $n\pm i$ , in order to cancel prime factors $p$ from the imaginary part; here $n$ is chosen to be a Størmer number such that $n^{2}+1$ is divisible by $p$ .[3]

References

Todd, John (1949), "A problem on arc tangent relations", American Mathematical Monthly, 56 (8): 517–528, doi:10.2307/2305526, JSTOR 2305526, MR 0031496.
Everest, Graham; Harman, Glyn (2008), "On primitive divisors of $n^{2}+b$ ", 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, MR 2428520. 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.

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[t-1] Todd, John (1949), "A problem on arc tangent relations", American Mathematical Monthly, 56 (8): 517–528, doi:10.2307/2305526, JSTOR 2305526, MR 0031496.

[eh-2] Everest, Graham; Harman, Glyn (2008), "On primitive divisors of $n^{2}+b$ ", 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, MR 2428520. See in particular Theorem 1.4 and Conjecture 1.5.

[cg-3] 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.