Størmer number
Encyclopedia
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
, etc. . Todd proved that this sequence is infinite (but not cofinite).
The Størmer numbers arise in connection with the problem of representing Gregory number
s ta / b as sums of Gregory numbers for integers: "To find Størmer's decomposition for ta / b, you repeatedly multiply a + bi by numbers n ± i for which n is a Størmer number and the sign is chosen so that you can cancel the corresponding prime number p (n is the smallest number for which n2 + 1 is divisible by p)."
Carl Størmer
Fredrik Carl Mülertz Størmer was a Norwegian mathematician and physicist, known both for his work in number theory and for studying the movement of charged particles in the magnetosphere and the formation of aurorae....
, 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
10 (number)
10 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...
, 11
11 (number)
11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...
, 12
12 (number)
12 is the natural number following 11 and preceding 13.The word "twelve" is the largest number with a single-morpheme name in English. Etymology suggests that "twelve" arises from the Germanic compound twalif "two-leftover", so a literal translation would yield "two remaining [after having ten...
, 14
14 (number)
14 is the natural number following 13 and preceding 15.In speech, the numbers 14 and 40 are often confused. When carefully enunciated, they differ in which syllable is stressed: 14 vs 40...
, 15
15 (number)
15 is the natural number following 14 and preceding 16. In English, it is the smallest natural number with seven letters in its spelled name....
, 16
16 (number)
16 is the natural number following 15 and preceding 17. 16 is a composite number, and a square number, being 42 = 4 × 4. It is the smallest number with exactly five divisors, its proper divisors being , , and ....
, 19
19 (number)
19 is the natural number following 18 and preceding 20. It is a prime number.In English speech, the numbers 19 and 90 are often confused. When carefully enunciated, they differ in which syllable is stressed: 19 vs 90...
, 20
20 (number)
20 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.-In mathematics:*20 is the basis for vigesimal number systems....
, etc. . Todd proved that this sequence is infinite (but not cofinite).
The Størmer numbers arise in connection with the problem of representing Gregory number
Gregory number
In mathematics, a Gregory number, named after James Gregory, is a real number of the form:where x is any rational number greater or equal to 1. Considering the power series expansion for arctangent, we have...
s ta / b as sums of Gregory numbers for integers: "To find Størmer's decomposition for ta / b, you repeatedly multiply a + bi by numbers n ± i for which n is a Størmer number and the sign is chosen so that you can cancel the corresponding prime number p (n is the smallest number for which n2 + 1 is divisible by p)."