Størmer number

A Stormer number , also known as Arc cotangent-irreducible number (English arc-cotangent irreducible number hereinafter) is a natural number , for which the largest prime factor of greater than or equal is. It is named after the Norwegian geophysicist and mathematician Carl Størmer . ${\displaystyle n}$ ${\displaystyle n^{2}+1}$ ${\displaystyle 2n}$

definition

A natural number is called a Størmer number if there is a prime with and , where | stands for the divisibility relation . ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle p\geq 2n}$ ${\displaystyle p|(n^{2}+1)}$

example

n = 33 is a Størmer number. The greatest prime factor of is , and it is greater than . ${\displaystyle n^{2}+1=33^{2}+1=1090=2\cdot 5\cdot 109}$ ${\displaystyle 109}$ ${\displaystyle 2n=2\cdot 33=66}$

Størmer numbers

The following numbers are Størmer numbers:

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, ...

John Todd has proven that this episode is neither finite nor finite .

Occur

Størmer numbers appear in integer places when examining values ​​of the arccotangent function. Such a value (also called Gregory number ) is called reducible if it is an integral linear combination ${\displaystyle \operatorname {arccot} n}$

${\displaystyle \operatorname {arccot} n=\sum _{k=1}^{n-1}a_{k}\cdot \operatorname {arccot} k,\quad a_{k}\in \mathbb {Z} }$

such values ​​can be written in smaller places, such as

${\displaystyle {\begin{array}{ll}\operatorname {arccot} 3&=\operatorname {arccot} 1-\operatorname {arccot} 2,\\\operatorname {arccot} 7&=\operatorname {arccot} 1-2\cdot \operatorname {arccot} 2,\\\operatorname {arccot} 8&=\operatorname {arccot} 3-\operatorname {arccot} 5,\\\operatorname {arccot} 57&=-2\cdot \operatorname {arccot} 1+3\cdot \operatorname {arccot} 2+\operatorname {arccot} 5,\\\operatorname {arccot} 239&=-\operatorname {arccot} 1+4\cdot \operatorname {arccot} 5,\\\operatorname {arccot} 682&=-2\cdot \operatorname {arccot} 1+3\cdot \operatorname {arccot} 2+2\cdot \operatorname {arccot} 11,\\\operatorname {arccot} 12943&=3\cdot \operatorname {arccot} 1-4\cdot \operatorname {arccot} 2-3\cdot \operatorname {arccot} 5+\operatorname {arccot} 11.\\\end{array}}}$

It turns out that it is irreducible, i.e. not such a linear combination, if and only if is a Størmer number. The type of decomposition shown explains the alternative term "arccotangent-irreducible number" mentioned at the beginning. ${\displaystyle \operatorname {arccot} n}$ ${\displaystyle n}$

Individual evidence

1. ^ John H. Conway , RK Guy : The Book of Numbers , Copernicus Press, p. 246
2. ↑ Follow A005528. Retrieved April 24, 2019 . 
3. ^ John Todd : A Problem on Arc Tangent Relations , American Mathematical Monthly (1949), Volume 56, No. 8, pp. 517-528. This also on JSTOR 2305526 .

Web links

• Størmer number (MathWorld)