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 Fredrik Carl Mülertz Størmer (3 September 1874 – 13 August 1957) was a Norwegian mathematician and astrophysicist. In mathematics, he is known for his work in number theory, including the calculation of and Størmer's theorem on consecu ...

Sequence

The first few Størmer numbers are:

Density

John Todd John Todd or Tod may refer to: Clergy *John Todd (abolitionist) (1818–1894), preacher and 'conductor' on the Underground Railroad * John Todd (author) (1800–1873), American minister and author * John Todd (bishop), Anglican bishop in the early ...

proved that this sequence is neither

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

nor

cofinite In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocoun ...

. More precisely, the

natural density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the de ...

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 The decimal value of the natural logarithm of 2 is approximately :\ln 2 \approx 0.693\,147\,180\,559\,945\,309\,417\,232\,121\,458. The logarithm of 2 in other bases is obtained with the formula :\log_b 2 = \frac. The common logarithm in particu ...

, approximately 0.693, but this remains unproven. 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 number In mathematics, a Gregory number, named after James Gregory, is a real number of the form: : G_x = \sum_^\infty (-1)^i \frac where ''x'' is any rational number greater or equal to 1. Considering the power series expansion for arctangent, we h ...

s (

arctangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...

s of

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

G_=\arctan\frac

as sums of Gregory numbers for integers (arctangents of

unit fraction A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, et ...

s). The Gregory number

G_

may be decomposed by repeatedly multiplying the

Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...

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

References

{{DEFAULTSORT:Stormer Number Integer sequences