Regular prime

Unsolved problem in mathematics

Are there infinitely many regular primes, and if so, is their relative density ${\displaystyle e^{-1/2}}$?

More unsolved problems in mathematics

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:

In 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent ${\displaystyle p}$ if ${\displaystyle p}$ is regular. This focused attention on the irregular primes.^[1] In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent ${\displaystyle p}$, if ${\displaystyle (p,p-3)}$ 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 ${\displaystyle (p,p-3)}$ or ${\displaystyle (p,p-5)}$ fails to be an irregular pair. (As applied in these results, ${\displaystyle (p,2k)}$ is an irregular pair when ${\displaystyle p}$ is irregular due to a certain condition, described below, being realized at ${\displaystyle 2k}$.)

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 ${\displaystyle (p,p-3)}$ is in fact an irregular pair for ${\displaystyle p=16843}$ and that this is the first and only time this occurs for ${\displaystyle p<30000}$.^[2] It was found in 1993 that the next time this happens is for ${\displaystyle p=2124679}$; see Wolstenholme prime.^[3]

An odd prime number ${\displaystyle p}$ is defined to be regular if it does not divide the class number of the ${\displaystyle p}$th cyclotomic field ${\displaystyle \mathbb {Q} (\zeta _{p})}$, where ${\displaystyle \zeta _{p}}$ is a primitive ${\displaystyle p}$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 ${\displaystyle \mathbb {Z} (\zeta _{p})}$ up to equivalence. Two ideals ${\displaystyle I}$ and ${\displaystyle J}$ are considered equivalent if there is a nonzero ${\displaystyle u}$ in ${\displaystyle \mathbb {Q} (\zeta _{p})}$ so that ${\displaystyle I=uJ}$. The first few of these class numbers are listed in (sequence A000927 in the OEIS).

Ernst Kummer (Kummer 1850) showed that an equivalent criterion for regularity is that ${\displaystyle p}$ does not divide the numerator of any of the Bernoulli numbers ${\displaystyle B_{k}}$ for ${\displaystyle k=2,4,6,\dots ,p-3}$.

Kummer's proof that this is equivalent to the class number definition is strengthened by the Herbrand–Ribet theorem, which states certain consequences of ${\displaystyle p}$ dividing the numerator of one of these Bernoulli numbers.

It has been conjectured that there are infinitely many regular primes. More precisely Carl Ludwig Siegel (1964) conjectured that ${\displaystyle e^{-1/2}}$, or about 60.65%, of all prime numbers are regular, in the asymptotic sense of natural density. Here, ${\displaystyle e\approx 2.718}$ is the base of the natural logarithm.

Taking Kummer's criterion, the chance that one numerator of the Bernoulli numbers ${\displaystyle B_{k}}$, ${\displaystyle k=2,\dots ,p-3}$, is not divisible by the prime ${\displaystyle p}$ is

$${\displaystyle {\dfrac {p-1}{p}}}$$

so that the chance that none of the numerators of these Bernoulli numbers are divisible by the prime ${\displaystyle p}$ is

$${\displaystyle \left({\dfrac {p-1}{p}}\right)^{\dfrac {p-3}{2}}=\left(1-{\dfrac {1}{p}}\right)^{\dfrac {p-3}{2}}=\left(1-{\dfrac {1}{p}}\right)^{-3/2}\cdot \left\lbrace \left(1-{\dfrac {1}{p}}\right)^{p}\right\rbrace ^{1/2}.}$$

By the definition of ${\displaystyle e}$, $${\displaystyle \lim _{p\to \infty }\left(1-{\dfrac {1}{p}}\right)^{p}={\dfrac {1}{e}}}$$ giving the probability $${\displaystyle \lim _{p\to \infty }\left(1-{\dfrac {1}{p}}\right)^{-3/2}\cdot \left\lbrace \left(1-{\dfrac {1}{p}}\right)^{p}\right\rbrace ^{1/2}=e^{-1/2}\approx 0.606531.}$$

It follows that about ${\displaystyle 60.6531\%}$ of the primes are regular by chance. Hart et al.^[4] indicate that ${\displaystyle 60.6590\%}$ of the primes less than ${\displaystyle 2^{31}=2,147,483,648}$ are regular.

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, ... (sequence A000928 in the OEIS)

K. L. Jensen (a student of Niels Nielsen^[5]) proved in 1915 that there are infinitely many irregular primes of the form ${\displaystyle 4n+3}$.^[6] In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes.^[7]

Metsänkylä proved in 1971 that for any integer ${\displaystyle T>6}$, there are infinitely many irregular primes not of the form ${\displaystyle mT\pm 1}$,^[8] and later generalized this.^[9]

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

The smallest even ${\displaystyle k}$ such that ${\displaystyle n}$th irregular prime divides ${\displaystyle B_{2k}}$ are

For a given prime ${\displaystyle p}$, the number of such pairs is called the index of irregularity of ${\displaystyle p}$.^[10] 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 ${\displaystyle (p,p-3)}$ is in fact an irregular pair for ${\displaystyle p=16843}$, as well as for ${\displaystyle p=2124679}$.. There are no more occurrences for ${\displaystyle p<10^{9}}$.

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

The irregular index of the ${\displaystyle n}$th prime starting with ${\displaystyle n=2}$, or the prime 3 is

The irregular index of the ${\displaystyle n}$th irregular prime is

The primes having irregular index 1 are

The primes having irregular index 2 are

The primes having irregular index 3 are

The least primes having irregular index ${\displaystyle n}$ are

(This sequence defines "the irregular index of 2" as −1, and also starts at ${\displaystyle n=-1}$.)

Similarly, we can define an Euler irregular prime (or E-irregular) as a prime ${\displaystyle p}$ that divides at least one Euler number ${\displaystyle E_{2n}}$ with ${\displaystyle 0<2n\leq p-3}$. The first few Euler irregular primes are

The Euler irregular pairs are

Vandiver proved in 1940 that Fermat's Last Theorem (that ${\displaystyle x^{p}+y^{p}=z^{p}}$ has no solution for integers ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$ with ${\displaystyle \gcd(xyz,p)=1}$) is true for prime exponents ${\displaystyle p}$ that are Euler-regular. Gut proved that ${\displaystyle x^{2p}+y^{2p}=z^{2p}}$ has no solution if ${\displaystyle p}$ has an E-irregularity index less than 5.^[11]

• Wolstenholme prime

• Weisstein, Eric W., "Irregular prime", MathWorld
• Chris Caldwell, The Prime Glossary: regular prime at The Prime Pages.
• Keith Conrad, Fermat's last theorem for regular primes.
• Bernoulli irregular prime
• Euler irregular prime
• Bernoulli and Euler irregular primes.
• Factorization of Bernoulli and Euler numbers
• Factorization of Bernoulli and Euler numbers