Regular prime
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In mathematics, regular primes are a certain kind of prime numbers. A regular prime p is one that does not divide the class number of the p-th cyclotomic field (that is, the algebraic number field obtained by adjoining the p-th root of unity to the rational numbers).
It can be shown that an equivalent criterion is that p does not divide the numerator of any of the Bernoulli numbers
- Bk
for
- k = 2, 4, 6, ..., p − 3.
Regular primes were first described by Ernst Kummer.
The first few regular primes are
It has been conjectured that there are infinitely many regular primes. More precisely it is expected that about
- e−½
of all prime numbers are regular, in the asymptotic sense of natural density (where e is the mathematical constant 2.71... ). Neither conjecture has been proven as of 2004.
Historically, regular primes were considered by Kummer since he was able to prove that Fermat's last theorem holds true for regular prime exponents (and consequently for all exponents that were multiples of regular primes).
A prime that is not regular is called an irregular prime; the number of Bk, the numerators of which p divides, is called the irregularity index of p.
It has been shown by Johan Jensen in 1915 that there are infinitely many irregular primes, the first few of which are
- 37, 59, 67, 101, 103, 131, 149, ... Template:OEIS.
External links
- The Prime Glossary: Regular prime (http://primes.utm.edu/glossary/page.php?sort=Regular)sl:regularno praštevilo