In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Størmer number or arc-cotangent irreducible number is a positive
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
for which the greatest
prime factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
of
is greater than or equal to
. 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 con ...
.
Sequence
The first Størmer numbers below 100 are:
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
Finite may refer to:
* 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 for person and/or tense or aspect
* "Finite", a song by Sara Gr ...
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 is countable, then one says the set is cocounta ...
.
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 desi ...
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
In mathematics, the natural logarithm of 2 is the unique real number argument such that the exponential function equals two. It appears frequently in various formulas and is also given by the alternating harmonic series. The decimal value of th ...
, 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 numbers (
arctangent
In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted domains. Specific ...
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 (for example,
The set of all ...
s)
as sums of Gregory numbers for integers (arctangents of
unit fraction
A unit fraction is a positive fraction with one as its numerator, 1/. It is the multiplicative inverse (reciprocal) of the denominator of the fraction, which must be a positive natural number. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. When a ...
s). The Gregory number
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 ...
by numbers of the form
, in order to cancel prime factors
from the imaginary part; here
is chosen to be a Størmer number such that
is divisible by
.
References
{{DEFAULTSORT:Stormer Number
Eponymous numbers in mathematics
Integer sequences