Citizendia
Your Ad Here

In number theory, a regular prime is a certain kind of prime number. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 A prime number p is called regular if it 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). In Mathematics, the extent to which Unique factorization fails in the ring of integers of an Algebraic number field (or more generally any Dedekind domain In Number theory, a cyclotomic field is a Number field obtained by adjoining a complex Root of unity to Q, the field of Rational numbers In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions Ernst Kummer showed that an equivalent criterion for regularity is that p does not divide the numerator of any of the Bernoulli numbers Bk for k = 2, 4, 6, …, p − 3. Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. Numerator may refer to A numeral used to indicate a count particularly of the equal parts in a fraction For example in 3/4 3 is the numerator In Mathematics, the Bernoulli numbers are a Sequence of Rational numbers with deep connections to Number theory. The first few regular primes are:

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, … (sequence A007703 in OEIS). ---- In mathematics Three is the first odd Prime number, and the second smallest prime This article discusses the number five. For the year 5 AD see 5. In mathematics Seven is the fourth Prime number. It is not only a Mersenne prime (since 23 &minus 1 = 7 but also a 17 ( seventeen) is the Natural number following 16 and preceding 18. 19 ( nineteen) is the Natural number following 18 and preceding 20. This article is about the number 23 For the year see 23. For the movies see 23 (film and The Number 23. 29 ( twenty-nine) is the Natural number following 28 and preceding 30. 31 ( thirty-one) is the Natural number following 30 and preceding 32. 41 ( forty-one) is the Natural number following 40 and preceding 42. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

It has been conjectured that there are infinitely many regular primes. In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness More precisely it is conjectured (Siegel, 1964) that e−1/2, or about 61%, of all prime numbers are regular, in the asymptotic sense of natural density. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line In pure and Applied mathematics, particularly the Analysis of algorithms, real analysis and engineering asymptotic analysis is a method of describing In Number theory, asymptotic density or natural density is one of the possibilities to measure how large is a Subset of the set of Natural Neither conjecture has been proven as of 2008.

Historically, regular primes were first considered by Kummer, who 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). Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like

An odd prime that is not regular is an irregular prime. The number of Bernoulli numbers Bk with a numerator divisible by p is called the irregularity index of p. K L Jensen has shown in 1915 that there are infinitely many irregular primes, the first few of which are:

37, 59, 67, 101, 103, 131, 149, … (sequence A000928 in OEIS). In Number theory, a regular prime is a certain kind of Prime number. 37 ( thirty-seven) is the Natural number following 36 and preceding 38. 59 ( fifty-nine) is the Natural number following 58 and preceding 60. 67 ( sixty-seven) is the Natural number following 66 and preceding 68. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

References

External links

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
 Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic