Regular prime
Encyclopedia
In number theory
, a regular prime is a prime number
p > 2 that does not divide the class number of the p-th cyclotomic field
.
The first few regular primes are:
showed that an equivalent criterion for regularity is that p does not divide the numerator of any of the Bernoulli number
s Bk for k = 2, 4, 6, …, p − 3.
d that there are infinitely many regular primes. More precisely Siegel conjectured (1964) that e
−1/2, or about 60.65%, of all prime numbers are regular, in the asymptotic
sense of natural density
. Neither conjecture has been proven .
K. L. Jensen has shown in 1915 that there are infinitely many irregular primes.
Metsänkylä proved that for any integer T > 6, there are infinitely many irregular primes not of the form mT + 1 or mT − 1.
For a given prime p, the number of such pairs is called the index of irregularity of p. Hence, a prime is regular if and only if its index of irregularity is zero. Similarly, a prime is irregular if and only if its index of irregularity is positive.
It was discovered that (p, p − 3) is in fact an irregular pair for p = 16843. This is the first and only time this occurs for p < 30000.
is true for a prime exponent p if p is regular. This raised attention in the irregular primes. In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent p, if is not an irregular pair. Kummer improved this further in 1857 by showing that for the first case of Fermat's Last Theorem it is sufficient to establish that either (p, p − 3) or (p, p − 5) fails to be an irregular pair.
Kummer found the irregular primes less than 165. In 1963, Lehmer reported results up to 10000 and Selfridge and Pollack announced in 1964 to have completed the table of irregular primes up to 25000. Although the two latter tables did not appear in print, Johnson found that is in fact an irregular pair for p = 16843 and that this is the first and only time this occurs for p < 30000.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a regular prime is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p > 2 that does not divide the class number of the p-th cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
.
The first few regular primes are:
- 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, ...
Kummer's criterion
Ernst KummerErnst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
showed that an equivalent criterion for regularity is that p does not divide the numerator of any of the Bernoulli number
Bernoulli number
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to number theory. They are closely related to the values of the Riemann zeta function at negative integers....
s Bk for k = 2, 4, 6, …, p − 3.
Properties
It has been conjectureConjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
d that there are infinitely many regular primes. More precisely Siegel conjectured (1964) that e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
−1/2, or about 60.65%, of all prime numbers are regular, in the asymptotic
Asymptotic analysis
In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The methodology has applications across science. Examples are...
sense of natural density
Natural density
In number theory, asymptotic density is one of the possibilities to measure how large a subset of the set of natural numbers is....
. Neither conjecture has been proven .
Irregular primes
An odd prime that is not regular is an irregular prime. The first few irregular primes are:- 37, 59, 67, 101, 103, 131, 149, ...
K. L. Jensen has shown in 1915 that there are infinitely many irregular primes.
Metsänkylä proved that for any integer T > 6, there are infinitely many irregular primes not of the form mT + 1 or mT − 1.
Irregular pairs
If p is an irregular prime and p divides the numerator of the Bernoulli number B2k for 0 < 2k < p − 1, then (p, 2k) is called an irregular pair. The first few irregular pairs are:- (691, 12), (3617, 16), (43867, 18), (283, 20), (617, 20), (131, 22), (593, 22), (103, 24), ... .
For a given prime p, the number of such pairs is called the index of irregularity of p. Hence, a prime is regular if and only if its index of irregularity is zero. Similarly, a prime is irregular if and only if its index of irregularity is positive.
It was discovered that (p, p − 3) is in fact an irregular pair for p = 16843. This is the first and only time this occurs for p < 30000.
History
In 1850, Kummer proved that Fermat's Last TheoremFermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
is true for a prime exponent p if p is regular. This raised attention in the irregular primes. In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent p, if is not an irregular pair. Kummer improved this further in 1857 by showing that for the first case of Fermat's Last Theorem it is sufficient to establish that either (p, p − 3) or (p, p − 5) fails to be an irregular pair.
Kummer found the irregular primes less than 165. In 1963, Lehmer reported results up to 10000 and Selfridge and Pollack announced in 1964 to have completed the table of irregular primes up to 25000. Although the two latter tables did not appear in print, Johnson found that is in fact an irregular pair for p = 16843 and that this is the first and only time this occurs for p < 30000.
Further reading
archived at WebCite archived at WebCiteExternal links
- Chris Caldwell, The Prime Glossary: regular prime at The Prime PagesPrime pagesThe Prime Pages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin.The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" lists for primes of various forms...
. - Keith Conrad, Fermat's last theorem for regular primes.