Regular prime

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In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers.

The first few regular odd primes are: Template:Bi

History and motivation

In 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent <math>p</math> if <math>p</math> is regular. This focused attention on the irregular primes.<ref name="Gardiner1988">Template:Citation</ref> In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent <math>p</math>, if <math>(p,p-3)</math> is not an irregular pair. Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem (see Sophie Germain's theorem) it is sufficient to establish that either <math>(p,p-3)</math> or <math>(p,p-5)</math> fails to be an irregular pair. (As applied in these results, Template:Nowrap is an irregular pair when <math>p</math> is irregular due to a certain condition, described below, being realized at <math>2k</math>.)

Kummer found the irregular primes smaller 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 <math>(p,p-3)</math> is in fact an irregular pair for <math>p=16843</math> and that this is the first and only time this occurs for <math>p<30000</math>.<ref>Template:Citation</ref> It was found in 1993 that the next time this happens is for <math>p=2124679</math>; see Wolstenholme prime.<ref>Template:Citation</ref>

Definition

Class number criterion

An odd prime number <math>p</math> is defined to be regular if it does not divide the class number of the <math>p</math>th cyclotomic field <math>\Q(\zeta_p)</math>, where <math>\zeta_p</math> is a primitive <math>p</math>th root of unity.

The prime number 2 is often considered regular as well.

The class number of the cyclotomic field is the number of ideals of the ring of integers <math>\Z(\zeta_p)</math> up to equivalence. Two ideals <math>I</math> and <math>J</math> are considered equivalent if there is a nonzero <math>u</math> in <math>\Q(\zeta_p)</math> so that <math>I=uJ</math>. The first few of these class numbers are listed in Template:Oeis.

Kummer's criterion

Template:Main Ernst Kummer Template:Harv showed that an equivalent criterion for regularity is that <math>p</math> does not divide the numerator of any of the Bernoulli numbers <math>B_k</math> for <math>k=2,4,6,\dots,p-3</math>.

Kummer's proof that this is equivalent to the class number definition is strengthened by the Herbrand–Ribet theorem, which states certain consequences of <math>p</math> dividing the numerator of one of these Bernoulli numbers.

Siegel's conjecture

It has been conjectured that there are infinitely many regular primes. More precisely Template:Harvs conjectured that <math>e^{-1/2}</math>, or about 60.65%, of all prime numbers are regular, in the asymptotic sense of natural density. Here, <math>e\approx 2.718</math> is the base of the natural logarithm.

Taking Kummer's criterion, the chance that one numerator of the Bernoulli numbers <math>B_k</math>, <math>k=2,\dots,p-3</math>, is not divisible by the prime <math>p</math> is

<math display=block>\dfrac{p-1}{p}</math>

so that the chance that none of the numerators of these Bernoulli numbers are divisible by the prime <math>p</math> is

<math display=block>\left(\dfrac{p-1}{p}\right)^{\dfrac{p-3}{2}}=\left(1-\dfrac{1}{p}\right)^{\dfrac{p-3}{2}}=\left(1-\dfrac{1}{p}\right)^{-3/2}\cdot\left\lbrace\left(1-\dfrac{1}{p}\right)^{p}\right\rbrace^{1/2}.</math>

By the definition of <math>e</math>, <math display=block>\lim_{p\to\infty}\left(1-\dfrac{1}{p}\right)^{p}=\dfrac{1}{e}</math> giving the probability <math display=block>\lim_{p\to\infty}\left(1-\dfrac{1}{p}\right)^{-3/2}\cdot\left\lbrace\left(1-\dfrac{1}{p}\right)^{p}\right\rbrace^{1/2}=e^{-1/2}\approx0.606531.</math>

It follows that about <math>60.6531\%</math> of the primes are regular by chance. Hart et al.<ref>Template:Citation</ref> indicate that <math>60.6590\%</math> of the primes less than <math>2^{31}=2,147,483,648</math> are regular.

Irregular primes

An odd prime that is not regular is an irregular prime (or Bernoulli irregular or B-irregular to distinguish from other types of irregularity discussed below). The first few irregular primes are:

37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593, ... Template:OEIS

Infinitude

K. L. Jensen (a student of Niels Nielsen<ref>Template:Citation</ref>) proved in 1915 that there are infinitely many irregular primes of the form <math>4n+3</math>.<ref>Template:Citation</ref> In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes.<ref>Template:Citation</ref>

Metsänkylä proved in 1971 that for any integer <math>T>6</math>, there are infinitely many irregular primes not of the form <math>mT\pm1</math>,<ref>Template:Citation</ref> and later generalized this.<ref>Template:Citation</ref>

Irregular pairs

If <math>p</math> is an irregular prime and <math>p</math> divides the numerator of the Bernoulli number <math>B_{2k}</math> for <math>0<2k<p-1</math>, then <math>(p,2k)</math> is called an irregular pair. In other words, an irregular pair is a bookkeeping device to record, for an irregular prime <math>p</math>, the particular indices of the Bernoulli numbers at which regularity fails. The first few irregular pairs (when ordered by <math>k</math>) are: Template:Bi

The smallest even <math>k</math> such that <math>n</math>th irregular prime divides <math>B_{2k}</math> are Template:Bi

For a given prime <math>p</math>, the number of such pairs is called the index of irregularity of <math>p</math>.<ref name=Nark475>Template:Citation</ref> 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 <math>(p,p-3)</math> is in fact an irregular pair for <math>p=16843</math>, as well as for <math>p=2124679</math>.. There are no more occurrences for <math>p<10^9</math>.

Irregular index

An odd prime <math>p</math> has irregular index <math>n</math> if and only if there are <math>n</math> values of <math>k</math> for which <math>p</math> divides <math>B_{2k}</math> and these <math>k</math>s are less than <math>(p-1)/2</math>. The first irregular prime with irregular index greater than 1 is 157, which divides <math>B_{62}</math> and <math>B_{110}</math>, so it has an irregular index 2. Clearly, the irregular index of a regular prime is 0.

The irregular index of the <math>n</math>th prime starting with <math>n=2</math>, or the prime 3 is Template:Bi

The irregular index of the <math>n</math>th irregular prime is Template:Bi

The primes having irregular index 1 are Template:Bi

The primes having irregular index 2 are Template:Bi

The primes having irregular index 3 are Template:Bi

The least primes having irregular index <math>n</math> are Template:Bi (This sequence defines "the irregular index of 2" as −1, and also starts at <math>n=-1</math>.)

Generalizations

Euler irregular primes

Similarly, we can define an Euler irregular prime (or E-irregular) as a prime <math>p</math> that divides at least one Euler number <math>E_{2n}</math> with <math>0<2n\le p-3</math>. The first few Euler irregular primes are Template:Bi

The Euler irregular pairs are Template:Bi

Vandiver proved in 1940 that Fermat's Last Theorem (that <math>x^p+y^p=z^p</math> has no solution for integers <math>x</math>, <math>y</math>, <math>z</math> with <math>\gcd(xyz,p)=1</math>) is true for prime exponents <math>p</math> that are Euler-regular. Gut proved that <math>x^{2p}+y^{2p}=z^{2p}</math> has no solution if <math>p</math> has an E-irregularity index less than 5.<ref>Template:Citation</ref>

See also

References

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Further reading

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