Regular prime number

In number theory , a prime number is called regular if it does not divide certain numbers . Its best-known application comes from Ernst Kummer , who proved in 1850 that Fermat's great theorem applies to exponents that are divisible by a regular prime number.

definition

A prime number is called regular if it does not share any of the numerators (in fully abbreviated form) of the Bernoulli numbers . ${\displaystyle p>2}$ ${\displaystyle B_{2},B_{4},\ldots ,B_{p-3}}$

In retrospect, Kummer showed that this is equivalent to the condition that the class number of the -th solid of the circle does not divide. ${\displaystyle p}$ ${\displaystyle p}$

Properties and things to know

A long open question is whether there are infinitely many regular prime numbers. Since sorrow the suspicion has been in the room that this is the case. It is further assumed that all prime numbers are regular. ${\displaystyle \mathrm {e} ^{-1/2}\approx 61\,\%}$

It is known that there are infinitely many irregular prime numbers ( theorem from KL Jensen 1915).

Regular prime numbers

The first members of the sequence are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, ... (sequence A007703 in OEIS ).

Irregular prime numbers

The first members of the sequence are 37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, ... (sequence A000928 in OEIS ).

Application to Fermat's great theorem: Sorrow's Theorem

The set of Kummer says:

The Fermat assumption is correct as long as the exponent in the Fermat's equation is a regular prime.

One possible proof of this is as follows:

Suppose is a regular prime, and it is with prime integers , with none of the figures by divisible is (this condition is called "case I"). Denotes a primitive -th root of unity, the left side of the equation can be factored as ${\displaystyle p}$ ${\displaystyle a^{p}+b^{p}=c^{p}}$ ${\displaystyle a,b,c}$ ${\displaystyle a,b,c}$ ${\displaystyle p}$ ${\displaystyle \zeta }$ ${\displaystyle p}$

${\displaystyle (a+b)(a+\zeta b)(a+\zeta ^{2}b)\cdots (a+\zeta ^{p-1}b),}$

and one can show that these factors are coprime pairs in the entirety ring . Since your product is a -th power, the individual factors are -th powers of ideals , especially so ${\displaystyle \mathbb {Z} [\zeta ]}$ ${\displaystyle c^{p}}$ ${\displaystyle p}$ ${\displaystyle p}$

${\displaystyle (a+\zeta b)={\mathfrak {a}}^{p}.}$

At this point the regularity of can now be used: The order of in the ideal class group can not divide because it has to divide the class number. However, the neutral element is in the ideal class group as is the main ideal. So the order of only 1 can be, itself is a main ideal. ${\displaystyle p}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle p}$ ${\displaystyle {\mathfrak {a}}^{p}}$ ${\displaystyle {\mathfrak {a}}^{p}}$ ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle {\mathfrak {a}}}$

That means: there is a unit and an element , so that ${\displaystyle u}$ ${\displaystyle t\in {\mathcal {O}}}$

${\displaystyle a+\zeta b=u\cdot t^{p}}$

applies.

This equation now leads modulo to contradiction on the way via congruence considerations . ${\displaystyle p}$

The set of sorrow is a milestone on the way of solving the Fermat problem . With the methods developed in the process, Kummer gave decisive impulses to later development .

literature

Original work

• EE Kummer : General proof of Fermat's theorem that the equation is unsolvable by integers, for all those power exponents that are odd prime numbers and do not occur as factors in the numerators of the first ½ Bernoulli numbers. ${\displaystyle x^{\lambda }+y^{\lambda }=z^{\lambda }}$ ${\displaystyle \lambda }$ ${\displaystyle (\lambda -3)}$In: Journal for Pure and Applied Mathematics ( Crelles Journal ). Volume 40, 1850, pp. 130-138 ( digizeitschriften.de ).
• KL Jensen: Om talteoretiske Egenskaber ved de Bernoulliske valley. In: Nyt Tidsskrift for Matematik. Afdeling B, Volume 26, 1915, ZDB -ID 281026-8 , pp. 73-83.

Monographs

• Peter Bundschuh: Introduction to Number Theory . 6th, revised and updated edition. Springer, Berlin et al. 2008, ISBN 978-3-540-76490-8 .
• Th. Skolem : Diophantine equations (= results of mathematics and its border areas. Volume 5, 4, ISSN  0071-1136 ). Springer, Berlin 1938 (Reprinted. Chelsea Publishing Company, New York NY 1950).

Web links

• Detailed proof of Fermat's large theorem for regular prime numbers (PDF; 137 kB, English)
• Original Evidence of Sorrow (Embedded PDF)
• Karl Dilcher: Bernoulli Bibliography. Dalhousie University, Halifax, Canada

Individual evidence

1. ↑ Bundschuh: p. 182.
2. ↑ Jensen: Nyt Tidskr. f. Math. Band 26 , p. 73 ff .
3. ↑ Bundschuh : p. 182.
4. ↑ Sorrow: Crelle's Journal . tape 40 , p. 130 ff .
5. ^ Skolem: p. 83.
6. ↑ Bundschuh: p. 182.