Regular prime: Difference between revisions
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==Definition== | ==Definition== | ||
A '''regular prime''' is a [[prime number]] greater than 2 such that { | A '''regular prime''' is a [[prime number]] greater than 2 such that <math>p</math> does '''not''' divide the class number of the cyclotomic number field <math>\mathbb{Q}(\zeta_p)</math>. | ||
A prime greater than 2 that is not a regular prime is termed an | A prime greater than 2 that is not a regular prime is termed an [[irregular prime]]. | ||
==Occurrence== | |||
===Initial examples=== | |||
<section begin="list"/>[[3]], [[5]], [[7]], [[11]], [[13]], [[17]], [[19]], [[23]], [[29]], [[31]], [[41]], [[43]], [[47]], [[53]], [[61]], <toggledisplay>[[71]], [[73]], [[79]], 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281, 313, 317, 331, 337, 349, 359, 367, 373, 383, 397, 419, 431</toggledisplay>[[Oeis:A007703|View list on OEIS]]<section end="list"/> | |||
==Facts== | ==Facts== | ||
* [[Infinitude conjecture for regular primes]]: It is conjectured that there are infinitely many regular primes, and in fact, the asymptotic density of regular primes is conjectured to be around <math>0.6</math>. | * [[Infinitude conjecture for regular primes]]: It is conjectured that there are infinitely many regular primes, and in fact, the asymptotic density of regular primes is conjectured to be around <math>0.6</math>. | ||
Latest revision as of 01:21, 3 July 2012
This article defines a property that can be evaluated for a prime number. In other words, every prime number either satisfies this property or does not satisfy this property.
View other properties of prime numbers | View other properties of natural numbers
Definition
A regular prime is a prime number greater than 2 such that does not divide the class number of the cyclotomic number field .
A prime greater than 2 that is not a regular prime is termed an irregular prime.
Occurrence
Initial examples
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, [SHOW MORE]
Facts
- Infinitude conjecture for regular primes: It is conjectured that there are infinitely many regular primes, and in fact, the asymptotic density of regular primes is conjectured to be around .
