Regular prime

In number theory, a prime number is called regular if it does not share specific numbers. Her best-known application is by Ernst Kummer, who proved in 1850 that the great Fermat's theorem is true for exponent, which are characterized by a regular prime number divisible.

• 4.1 Original Articles
• 4.2 monographs

Definition

A prime number is called regular if it does not share the counter ( in a fully shortened representation ) of the Bernoulli numbers.

Grief hindsight shows that this is equivalent to the condition that does not share the class number of the -th cyclotomic field.

Features and facts

A long time open question is whether there are infinitely many regular primes. Grief since the assumption is in the room that is the case. It is assumed further that all primes are regular.

It is known that there are infinitely many irregular primes ( set of KL Jensen 1915). The smallest irregular prime numbers are 37, 59, 67

Regular primes

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

Irregular primes

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

Application to the large set of Fermat: The set of sorrow

The set of sorrow says:

A possible proof of this is as follows:

Suppose is a regular prime number, and it is with coprime integers such that none of the numbers divisible by (this condition is called " Case I "). Describes a primitive root of unity, so can factor the left side of the equation as

And one can show that these factors are pairwise relatively prime in the whole ring. Since their product is a -th power, and the individual factors - te are powers of ideals, ie in particular

At this point, the regularity can now be used by: the order of the ideal class group can not share, since it must be divisor of the class number. However, the neutral element in the ideal class group, as principal ideal. So, the order of only 1 be, itself is a principal ideal.

This means that there is a unit, and an element, so that

Applies.

This equation now leads by way of Kongruenzbetrachtungen modulo a contradiction.

The set of grief is a milestone on the way to solving the Fermat problem. Through the methods developed grief has given the later development decisive impulses.