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Regular prime

In number theory, a regular prime is an odd prime number p such that p does not divide the class number h of the p-th cyclotomic field \mathbb{Q}(\zeta_p), where \zeta_p is a primitive p-th root of unity.^[1] Equivalently, p is regular if it does not divide the numerator of any even-indexed Bernoulli number B_k for $2 \leq k \leq p-3, when these fractions are expressed in lowest terms with denominator a power of 2.^[2] This equivalence, established by Ernst Kummer in 1850, links the arithmetic of cyclotomic fields to properties of Bernoulli numbers via the von Staudt–Clausen theorem and Kummer's congruence relations.^[2] The concept of regular primes was introduced by Kummer in the mid-19th century as part of his groundbreaking work on Fermat's Last Theorem (FLT), which conjectures that there are no positive integers a, b, c satisfying a^n + b^n = c^n for n > 2.^[1] Motivated by Gabriel Lamé's flawed 1847 attempt to prove FLT using unique factorization in cyclotomic rings, Kummer developed the theory of ideal numbers to restore unique factorization in these rings for regular primes.^[1] Using this framework, he proved FLT for all regular primes p in both Case I (where p does not divide abc) and Case II (where p divides c but not a or b), covering a significant portion of odd primes and advancing the theorem's understanding until Andrew Wiles' complete proof in 1994.^[1] Among the first few odd primes, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31 are regular, while 37 is the smallest irregular prime.^[3] Up to 100, there are 21 regular primes out of 24 odd primes, with known irregular ones including 37, 59, and 67.^[3] It is relatively straightforward to prove that there are infinitely many irregular primes, but the existence of infinitely many regular primes remains an open conjecture, supported by heuristic arguments based on the distribution of Bernoulli number numerators modulo p. Computational evidence, such as the determination of all irregular primes up to 2 billion (Kraus, 2016), supports the heuristic that approximately 60% of primes are regular.^[3]^[4]

History and motivation

Kummer's foundational work

In the mid-19th century, Ernst Kummer investigated the failure of unique factorization in the ring of integers of cyclotomic fields \mathbb{Q}(\zeta_p), where \zeta_p is a primitive p-th root of unity and p is an odd prime. This issue became evident when Gabriel Lamé announced an incorrect proof of Fermat's Last Theorem in 1847, relying on the assumption that unique factorization holds in \mathbb{Z}[\zeta_p], which Kummer demonstrated was false for certain primes, such as p=23. To address this, Kummer developed the theory of ideal numbers starting in 1844, a precursor to Dedekind's ideals, allowing him to restore a form of unique factorization for ideals in these rings.^[1]^[5] Kummer introduced the concept of regular primes in 1850 as part of his efforts to apply this ideal theory to Fermat's Last Theorem. A prime p was deemed regular if the class number of \mathbb{Q}(\zeta_p) was not divisible by p, ensuring that certain ideal factorizations behaved sufficiently well for his proofs. This innovation was detailed in his papers on the class numbers of cyclotomic fields, published in 1850 and 1851 in the Journal für die reine und angewandte Mathematik.^[1] The primary motivation for defining regular primes stemmed from Kummer's attempt to prove Fermat's Last Theorem using infinite descent methods in the cyclotomic integers. By factoring equations like x^p + y^p = z^p into ideals in \mathbb{Z}[\zeta_p] and leveraging the regularity condition, Kummer could show that any supposed solution would lead to a contradiction through descent, provided p was regular. This approach succeeded where Lamé's had failed, by accounting for the class group structure.^[5]^[6] Kummer explicitly proved that Fermat's Last Theorem holds for all regular odd primes p < 100, excluding the irregular cases p = 37, 59, 67, where the class number is divisible by p. His computations confirmed that 21 of the 24 odd primes less than 100 are regular, covering a significant portion of small exponents and marking a major advance in the theorem's partial resolution. Irregular primes, the complement, thus became the focus of further study.^[5]^[1]

Connection to Fermat's Last Theorem

Kummer's approach to proving Fermat's Last Theorem (FLT) for prime exponents relied on an infinite descent argument within the cyclotomic field \mathbb{Q}(\zeta_p), where \zeta_p is a primitive p-th root of unity. This descent uses ideal factorization in the ring of integers \mathbb{Z}[\zeta_p] via Kummer's theory of ideal numbers; regularity (where p does not divide the class number h_p) ensures that the p-primary component of the class group is trivial, allowing the necessary steps in the argument to proceed without obstruction from non-principal ideals. Irregular primes, where p divides h_p, introduce complications in the class group that prevent the descent from working directly.^[5] In 1847, Kummer established FLT for all regular prime exponents p, demonstrating that no nontrivial solutions exist to x^p + y^p = z^p in positive integers under this condition. This covered a significant portion of cases, as the first few odd primes (3, 5, 7, 11, 13, 17, 19, 23, 29, 31) are all regular, allowing Kummer's method to verify FLT for these small exponents. However, the smallest irregular prime, 37, halted the approach, as the class number of \mathbb{Q}(\zeta_{37}) is divisible by 37, rendering the descent invalid.^[1]^[7] Subsequent advancements extended Kummer's results to certain irregular primes. In the 1920s, Philipp Furtwängler utilized class field theory and higher reciprocity laws to prove FLT for some irregular cases like 37, 59, and 67, by developing methods to handle mild irregularities in the class group structure. Later, in 1955, H. S. Vandiver proved FLT for all prime exponents less than 4002, including remaining irregular primes up to that bound, through detailed computations of class numbers. These partial extensions highlighted the limitations of regularity but paved the way for broader strategies.^[8]^[9] Andrew Wiles' 1995 proof of FLT employed modular forms and elliptic curves, bypassing cyclotomic fields entirely, yet the study of regular primes indirectly informed this modular approach by emphasizing the role of Galois representations and arithmetic structures in Diophantine equations. Historical insights from Kummer's work on ideal class groups influenced the development of the Langlands program, which underpinned Wiles' strategy of linking semistable elliptic curves to modular forms.^[9]^[10]

Definition

Class number criterion

The p-th cyclotomic field for an odd prime p is the extension \mathbb{Q}(\zeta_p) of the rational numbers \mathbb{Q} obtained by adjoining a primitive p-th root of unity \zeta_p = e^{2\pi i / p}. This is a Galois extension of degree p-1 with abelian Galois group isomorphic to (\mathbb{Z}/p\mathbb{Z})^\times. The ring of integers of \mathbb{Q}(\zeta_p) is \mathbb{Z}[\zeta_p], a monogenic Dedekind domain in which every nonzero ideal factors uniquely into prime ideals.^[5] The class number h_p of \mathbb{Q}(\zeta_p) is the cardinality of its ideal class group, which classifies fractional ideals up to principal ideals and quantifies the extent to which unique factorization into elements fails in \mathbb{Z}[\zeta_p]. If h_p = 1, then \mathbb{Z}[\zeta_p] is a principal ideal domain, so every ideal is principal and algebraic integers factor uniquely into irreducibles up to units.^[1] A prime p is regular if and only if p does not divide h_p, meaning the ideal class group of \mathbb{Q}(\zeta_p) has no p-torsion (i.e., its p-Sylow subgroup is trivial). Equivalently, h_p is coprime to p. This formulation, introduced by Ernst Kummer in his studies on Fermat's Last Theorem, serves as the primary class number criterion for regularity.^[5]^[1] For regular primes, the condition p \nmid h_p ensures that the prime p ramifies in a way that avoids complications in the p-primary component of the class group, enabling algebraic techniques reliant on ideal factorization without interference from non-principal ideals divisible by p. This property was essential for Kummer's proofs of Fermat's Last Theorem in the case x^p + y^p = z^p with no positive integer solutions, as it allows descent arguments using principal ideals in \mathbb{Z}[\zeta_p].^[5]^[1]

Equivalent criteria

A prime p is regular if and only if the relative class number h_p^- of the p-th cyclotomic field \mathbb{Q}(\zeta_p) is not divisible by p. The class number h_p factors as h_p = h_p^+ \cdot h_p^-, where h_p^+ is the class number of the maximal real subfield \mathbb{Q}(\zeta_p + \zeta_p^{-1}). The factor h_p^+ is computed via ideal class group algorithms involving factorization of prime ideals of small norm in the real subfield, and its value is not divisible by p due to bounds on class numbers relative to the degree and discriminant. Thus, regularity reduces to checking the p-divisibility of h_p^-, which can be computed using methods such as the Stickelberger relations acting on the class group or evaluations of p-adic L-functions.^[11] Another classical equivalent criterion involves Bernoulli numbers: p is regular if and only if p does not divide the numerator of the Bernoulli number B_k (expressed in lowest terms) for any even integer k with $2 \leq k \leq p-3. This equivalence follows from Kummer's congruences relating the numerators of Bernoulli numbers to the p-adic behavior of the class group, combined with the von Staudt–Clausen theorem describing the denominators of Bernoulli numbers.^[2] The Herbrand-Ribet theorem provides an equivalent criterion in terms of Galois representations: p is irregular if and only if the \chi^{1-k}-isotypical component of the class group of \mathbb{Q}(\zeta_p) is nontrivial for some even integer k with $2 \leq k \leq p-3, where \chi is the cyclotomic character; this is equivalent to the nonvanishing of certain p-adic L-functions or the existence of associated modular forms with specific properties. This reformulation connects the algebraic condition on the class number to analytic and modular objects.^[12]

Theoretical results

Siegel's theorem

In the 1960s, Carl Ludwig Siegel developed key results on the size of class numbers in cyclotomic fields, providing asymptotic bounds that have significant implications for the study of regular primes. Siegel's work built on earlier efforts to understand the growth of class numbers in number fields of increasing degree, addressing the ineffective aspects of previous estimates by leveraging analytic methods from the theory of L-functions. His theorems offered the first unconditional upper bounds on the class number of the p-th cyclotomic field \mathbb{Q}(\zeta_p), resolving partial ineffectiveness in bounds for the product of the class number and regulator.^[13] A central result is Siegel's ineffective upper bound on the logarithm of the class number h_p of \mathbb{Q}(\zeta_p): for any \varepsilon > 0, \log h_p = o(p^{1/2 + \varepsilon}). This bound is ineffective because the implied constant depends on \varepsilon in a way that cannot be explicitly computed, stemming from the potential existence of exceptional zeros (Siegel zeros) in associated Dirichlet L-functions. The theorem can be stated more precisely as h_p < C_\varepsilon p^{1/2 + \varepsilon} for some constant C_\varepsilon > 0, where the constant is ineffective. This follows from applying analytic estimates to the residue of the Dedekind zeta function at s=1, incorporating the class number formula and bounds on L(1, χ) for characters mod p. Siegel's bound implies that there are only finitely many primes p with h_p = 1, specifically h_p = 1 only for p bounded by some ineffective constant. Under the generalized Riemann hypothesis (GRH), the bound becomes effective, allowing an explicit (though very large) threshold beyond which h_p > 1. Without GRH, the ineffectiveness prevents a computable threshold, but the result provides strong evidence that the class number grows relative to the degree of the field. For regular primes, which are those for which p does not divide the class number h_p of \mathbb{Q}(\zeta_p) (equivalently, h_p^+ of the maximal real subfield \mathbb{Q}(\zeta_p)^+, since h_p = h_p^+), this bound suggests that the p-primary part of h_p is trivial for "large" p in a heuristic sense, as the overall size of h_p is constrained relative to p, though ineffectiveness limits proofs. Thus, it supports the conjecture that a positive proportion of primes are regular (with natural density $1/\sqrt{e} \approx 0.6065), though it falls short of proving the infinitude of regular primes due to the ineffective constant.

Density of regular primes

Siegel conjectured in 1964 that the set of regular primes has positive natural density \frac{1}{\sqrt{e}} \approx 0.6065 among all primes. This heuristic arises from probabilistic models assuming the numerators of the relevant Bernoulli numbers behave randomly modulo p, leading to the expected proportion of primes dividing the class number being $1 - \frac{1}{\sqrt{e}} \approx 0.3935. No unconditional proof establishes even the infinitude of regular primes, let alone a positive density. Under the generalized Riemann hypothesis (GRH), stronger results on the distribution of primes in certain arithmetic progressions exist, but they do not yet yield a positive density for regular primes without additional assumptions. Siegel's earlier theorem on effective bounds for class numbers provides foundational estimates supporting the plausibility of his density conjecture, though it falls short of proving it. Computational evidence aligns closely with Siegel's predicted proportion. Samuel S. Wagstaff Jr. determined all irregular primes up to 125,000, revealing that 60.75% of primes in this range are regular. Extending to larger bounds, computations up to $4 \times 10^6 show approximately 60.59% regular primes, with the proportion of irregular primes at 39.41%, remarkably near the conjectured 39.35%. Further computations up to 12 million (as of 2000) confirm a proportion of about 60% regular primes, with the estimate stabilizing near Siegel's value.^[14]

Irregular primes

Definition and basic properties

An odd prime p is irregular if it divides the class number h_p of the pth cyclotomic field \mathbb{Q}(\zeta_p), where \zeta_p is a primitive pth root of unity; equivalently, this occurs if and only if p divides the numerator of at least one Bernoulli number B_k for even k with $2 \leq k \leq p-3.^[15] This contrasts with regular primes, for which p does not divide h_p (and hence no such Bernoulli numerators). The smallest irregular primes are 37, 59, 67, and 101. For p=37, the class number is h_{37}=37.^[16] For p=59, h_{59}=41241=59 \times 699. Computations show that up to $10^6, there are approximately 39% irregular primes among the odd primes.^[17] Irregularity implies that the ring of integers \mathbb{Z}[\zeta_p] of \mathbb{Q}(\zeta_p) has class number greater than 1, so unique factorization into prime ideals fails.^[18] In particular, no regular prime divides the numerator of B_{p-3}, but irregular primes may do so for that or earlier Bernoulli numbers in the range. Irregular primes are also connected to failures of Leopoldt's conjecture in certain non-abelian extensions, where the p-adic regulator defect relates to the nontrivial p-part of the class group.^[18]

Infinitude and distribution

The existence of infinitely many irregular primes was established by K. L. Jensen in 1915, who demonstrated that there are infinitely many such primes congruent to 3 modulo 4. Jensen's proof relies on genus theory for binary quadratic forms and the growth of class numbers in imaginary quadratic fields, showing that for sufficiently many primes p ≡ 3 mod 4, the prime p divides the class number of the cyclotomic field \mathbb{Q}(\zeta_p), thereby rendering p irregular. A simpler alternative proof was later provided by L. Carlitz in 1954, which proceeds by contradiction: assuming only finitely many irregular primes p_1, \dots, p_k, one considers the Bernoulli number B_m where m = \prod_{i=1}^k (p_i - 1); the rapid asymptotic growth |B_{2\ell}/(2\ell)| \sim 2 (2\ell)! / (2\pi)^{2\ell} as \ell \to \infty implies that the numerator of B_m / m must be divisible by some new prime q > all p_i, and since m < q-1, this q is irregular, yielding the contradiction.^[19] Regarding distribution, it is conjectured that the irregular primes have positive relative density $1 - e^{-1/2} \approx 0.3935 among all primes, a heuristic originally proposed by C. L. Siegel in 1964 based on the expected number of Bernoulli numerators divisible by p being approximately (p-3)/(2p) \approx 1/2, leading to the probability of no such divisions being e^{-1/2} for regularity. Under the generalized Riemann hypothesis (GRH), this density is known to exist and equals the conjectured value, as the non-vanishing of certain L-functions at s=1 ensures the precise distribution of the relevant arithmetic progressions and congruences modulo p. Unconditionally, stronger quantitative bounds on the counting function have been established; for instance, the number I(x) of irregular primes up to x satisfies I(x) \gg \frac{\log x \cdot \log \log \log x}{\log \log x}, implying that the lower asymptotic density is positive in a logarithmic sense, though the exact lim inf I(x)/\pi(x) > 0 remains tied to GRH for the full proportion. This growth ties into broader analytic number theory, particularly the non-vanishing properties of Dirichlet L-functions associated with cyclotomic characters.^[20] Computational evidence supports the conjectured density, with the proportion of irregular primes stabilizing around 39%. For example, exhaustive searches up to 163,577,356 identify 3,604,564 irregular primes, yielding a relative density of approximately 0.393. More recent computations up to $2^{31} = 2,147,483,648 (as of 2016) show approximately 41,346,000 irregular primes out of 105,097,564 odd primes, with a density of about 0.3934, consistent with the conjecture and confirming no significant deviations.^[21]^[22]

Irregular pairs and index

For an irregular prime p, an irregular pair is a pair (p, 2k) where k is an integer with $1 \le k \le (p-3)/2, and p divides the numerator of the Bernoulli number B_{2k}. This condition is equivalent to p dividing the class number h_p of the p-th cyclotomic field \mathbb{Q}(\zeta_p), with the specific $2k indicating the contribution to the irregularity. The exponent e in such pairs can be extended to higher powers, where p^e divides B_{2k}/2k, but the basic pair focuses on e \ge 1.^[23] The irregular index i_p, also known as the index of irregularity, is defined as the number of distinct irregular pairs (p, 2k) for a given irregular prime p, providing a measure of the degree of irregularity. A prime p is regular if i_p = 0, and irregular otherwise. This index quantifies how many Bernoulli numbers contribute to the failure of regularity, and it is closely related to the structure of the p-class group of \mathbb{Q}(\zeta_p), where the p-rank equals i_p by the Mazur-Wiles theorem.^[24] Statistical studies show that most irregular primes have irregular index 1, with higher indices becoming progressively rarer, following a Poisson distribution with mean approximately $1/2. For example, the irregular prime p = 37 has index 1, corresponding to the irregular pair (37, 32), where 37 divides the numerator of B_{32}. Another example is p = 1327, which has index 2, indicating two distinct irregular pairs contributing to its irregularity. Among irregular primes below 125,000, approximately 77% have index 1, 19% have index 2, and higher indices occur in less than 4% of cases.^[24]^[23] The irregular index plays a key role in analyzing the distribution of irregular primes and in developing effective versions of class number problems. For instance, it aids in bounding the p-part of h_p and verifying criteria for Fermat's Last Theorem in cyclotomic rings, as higher indices complicate the vanishing of certain cyclotomic units. Computations of the index also support conjectures on the density of regular primes, estimated at about 60.65% asymptotically.^[24]

Generalizations

Euler irregular primes

In the context of algebraic number theory, Euler irregular primes represent a generalization of Kummer's irregular primes, where the role of Bernoulli numbers is played by Euler numbers E_{2n}, the coefficients in the Taylor series expansion of \sech x = \frac{2}{e^x + e^{-x}}.^[25] A prime p > 2 is defined as Euler-irregular if it divides at least one Euler number E_{2n} for $0 < 2n < p-1. The Euler numbers are integers satisfying E_0 = 1, E_2 = -1, E_4 = 5, E_6 = -61, E_8 = 1385, and so on, with alternating signs and rapid growth in magnitude. This condition parallels the divisibility criterion for Kummer's irregular primes but arises in studies of congruences involving secant and tangent functions rather than those for the Riemann zeta function. The notion first gained prominence in H. S. Vandiver's 1940 investigations into Fermat's Last Theorem, where he established that if the equation x^p + y^p = z^p admits a nontrivial solution in integers coprime to p, then p must divide E_{p-3}.^[26] Subsequent work by I. Gut in 1950 and L. Carlitz in 1954 further developed the theory, with Carlitz providing a proof of the infinitude of such primes. Unlike Kummer's irregular primes, which relate directly to the class number of the p-th cyclotomic field, Euler irregular primes connect to properties of the real cyclotomic field and have been linked to Wieferich-like congruences for Euler numbers. For instance, primes that divide E_{p-3} form a subclass of interest for potential counterexamples to Fermat's Last Theorem, though none are known beyond trivial cases since the theorem holds. Computations up to large bounds reveal that Euler irregular primes are relatively sparse among all primes; the smallest examples include 19 (dividing E_{10} = -50521), 31, 43 (dividing E_{20}), 47, 61, 67, 71, 79, and 101. Notably, some primes like 19 are regular in the Kummer sense but Euler-irregular, highlighting the distinct nature of the criteria.^[25] Theoretical results establish the existence of infinitely many Euler irregular primes, as proved by Carlitz using properties of the distribution of prime factors in Euler numbers and Dirichlet's theorem on primes in arithmetic progressions. Additionally, there are infinitely many such primes congruent to 1 modulo 8, due to R. Ernvall's 1975 analysis of residue classes. Heuristics based on a Poisson distribution model for the independence of divisibility events across Euler numbers suggest that approximately 39.35% of all primes are Euler-irregular, though this remains conjectural and aligns with computational evidence up to $10^{12}. Unlike Kummer irregular primes, where the density is expected to approach 1, the Euler variant exhibits a positive but sub-unity asymptotic density under these probabilistic assumptions.^[25]

Broader extensions

The notion of regular primes has been extended to higher-degree cyclotomic fields \mathbb{Q}(\zeta_{p^k}) for k > 1, where analogous regularity conditions focus on the p-part of the class number remaining undivided by p across layers of the cyclotomic \mathbb{Z}_p-extension. In this tower, the class number growth is governed by Iwasawa invariants \mu, \lambda, and \nu, with the formula \log_p \# H_n = \mu p^n + \lambda n + \nu for sufficiently large n, where H_n is the p-class group of the nth layer. For the cyclotomic \mathbb{Z}_p-extension over \mathbb{Q}, \mu = 0 holds unconditionally, reflecting a form of regularity in the absence of exponential growth in the p-class number; this aligns with Kummer's original condition at k=1, as irregular primes contribute to positive \lambda values in higher layers.^[27] In broader abelian extensions, such as real quadratic fields k_0 = \mathbb{Q}(\sqrt{d}) for square-free d > 0, the concept of regularity is generalized using values of the Dedekind zeta function \zeta_{k_0}(s) at negative integers, analogous to Bernoulli numbers in the cyclotomic case. A prime p is k_0-regular if p does not divide the numerators of these zeta values for $1 \leq m \leq p-3, and the k_0-irregularity index measures the extent of deviation, similar to Kummer's index for irregular primes. Computational studies show that the distribution of these indices follows heuristic predictions, with about 40% of primes being irregular for small real quadratic fields like \mathbb{Q}(\sqrt{5}). For fields with class number 1, such as the nine known imaginary quadratic fields, this principal ideal property implies analogs of regularity by ensuring unique factorization in the ring of integers, facilitating splitting behaviors akin to those in cyclotomic settings.^[28]^[29] p-adic notions of regularity arise in Iwasawa theory through the \mu-invariant, where \mu = 0 signifies "regular" behavior in \mathbb{Z}_p-extensions, preventing pathological exponential growth in class numbers. This condition is verified for the cyclotomic \mathbb{Z}_p-extension of abelian extensions of \mathbb{Q}, tying back to regular primes via the main conjecture, which equates the characteristic ideal of the Iwasawa module to a p-adic L-function; for irregular primes like 37, \lambda > 0 captures the nontrivial p-torsion in the base layer's class group. In CM fields, which are totally imaginary quadratic extensions of totally real fields, class number 1 (as in the Heegner list) provides regularity analogs by ensuring the ring of integers is a PID, with implications for prime splitting and endomorphism rings in associated elliptic curves.^[27]^[30] Post-2000 developments link regular primes to the Langlands program through the distribution of class numbers in cyclotomic fields, interpreted via motives and automorphic forms; however, while heuristic models suggest connections to motive realizations over number fields, no major breakthroughs tying irregularity directly to Langlands reciprocity have emerged by 2025.^[31]

References

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### Definitions and Information
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