Størmer number

In mathematics, a Størmer number or arc-cotangent irreducible number, named after Carl Størmer, is a positive integer n for which the greatest prime factor of (n² + 1) meets or exceeds 2n.

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).

J. Todd proved that this sequence is neither finite nor cofinite.

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 $p$ .^[1]

Notes

↑ Conway & Guy (1996): 245, ¶ 3

References

John H. Conway & R. K. Guy, The Book of Numbers. New York: Copernicus Press (1996): 245–248.
J. Todd, "A problem on arc tangent relations", Amer. Math. Monthly, 56 (1949): 517–528.

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
via a sieve

Lucky

Code related

Meertens

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star

non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral

non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula

pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular

non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

By properties of Ω(n)	Almost prime Semiprime

By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient

By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
numbers

Recreational
mathematics

Base-dependent
numbers

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