Størmer number

{{Short description|Number n where the highest prime factor of (n^2 + 1) is at least 2n}}

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 Størmer numbers below 100 are:

{{bi|left=1.6| 1, 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, 51, 52, 53, 54, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 74, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 92, 94, 95, 96, 97... {{OEIS|id=A005528}}. }}

The only numbers below 100 that aren't Størmer are 3, 7, 8, 13, 17, 18, 21, 30, 31, 32, 38, 41, 43, 46, 47, 50, 55, 57, 68, 70, 72, 73, 75, 76, 83, 91, 93, 98, 99 and 100.

Density

John Todd proved that this sequence is neither finite nor cofinite.{{r|t}}

{{unsolved|mathematics|What is the natural density of the Størmer numbers?}}

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.{{r|eh}}

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 $p$ .{{r|cg}}

References

{{reflist|refs=

{{citation

| last1 = Conway | first1 = John H. | author1-link = John Horton Conway

| last2 = Guy | first2 = R. K. | author2-link = Richard K. Guy

| location = New York

| pages = 245–248

| publisher = Copernicus Press

| title = The Book of Numbers

| year = 1996}}. See in particular p. 245, para. 3.

{{citation

| last1 = Everest | first1 = Graham

| last2 = Harman | first2 = Glyn

| arxiv = math/0701234

| contribution = On primitive divisors of $n^2+b$

| doi = 10.1017/CBO9780511721274.011

| mr = 2428520

| pages = 142–154

| publisher = Cambridge Univ. Press, Cambridge

| series = London Math. Soc. Lecture Note Ser.

| title = Number theory and polynomials

| volume = 352

| year = 2008}}. See in particular Theorem 1.4 and Conjecture 1.5.

{{citation

| last = Todd | first = John | authorlink = John Todd (computer scientist)

| doi = 10.2307/2305526

| journal = American Mathematical Monthly

| mr = 0031496

| pages = 517–528

| title = A problem on arc tangent relations

| volume = 56

| year = 1949| issue = 8 | jstor = 2305526 }}.

}}

Category:Eponymous numbers in mathematics

Category:Integer sequences