##  [Irregular Prime](/irregular-prime-0) 

 Definition

A prime p that divides the numerator of some Bernoulli number B_k (typically for even k with 2 ≤ k ≤ p−3), so p fails Kummer's regularity condition and introduces complications in class groups of cyclotomic fields and related cyclotomic arithmetic.

 

 

 

 

 

 





## Principle

Principle

Kummer defined a prime p to be regular if it does not divide the numerator of any Bernoulli number B_{2k} with 2k between 2 and p−3; an irregular prime violates this, and such divisibilities are tied to nontrivial p-torsion in class groups of cyclotomic fields and obstructions in cyclotomic Iwasawa theory.

 

 

 

 

 





## Demonstration

Demonstration

If p divides the numerator of B_{2k} for some 2k with 2 ≤ 2k ≤ p−3, then p is irregular; this congruence appears in the study of the structure of the p-part of the class group of Q(ζ_p) and shows up in Kummer's investigations of ideal class groups and Bernoulli numerators.

 

 

 

 

## Misapplication

Misapplication

Confusing irregular primes with primes having other exceptional properties (e.g., primitive root or Wieferich behavior) or assuming every irregular prime necessarily yields a failure of well-known conjectures — irregularity is a precise Bernoulli divisibility condition and has specific arithmetic consequences, not universal pathology.

 

 

 

 

 





## Consequence

Consequence

Irregular primes complicate the arithmetic of cyclotomic fields, can cause the presence of nontrivial p-torsion in class groups, affect the validity of certain cyclotomic criteria (as used in Kummer's approach to Fermat's Last Theorem historically) and play a central role in Iwasawa-theoretic phenomena.

 

 

 

 

## Reversal

Reversal

Regular prime: p does not divide any relevant Bernoulli numerator, so Kummer-type arguments about cyclotomic class groups are simpler and certain divisibility-based obstructions are absent.

 

 

 

 

 





## Boundary

Boundary

Applies to odd primes in relation to Bernoulli numbers and cyclotomic fields; 2 is typically excluded by the usual definition, and one must specify which Bernoulli indices are considered (the classical range 2 ≤ 2k ≤ p−3).

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension arises between the elementary divisibility condition on Bernoulli numerators and deeper manifestations in class groups and Iwasawa invariants — the same divisibility criterion can be viewed as an explicit congruence or as a signal of subtler arithmetic structure.

 

 

 

 

 





## Synthesis

Synthesis

An irregular prime is exactly a prime that divides the numerator of some Bernoulli number in the classical index range; this simple divisibility encodes nontrivial information about p-torsion in cyclotomic class groups and is a gateway to more elaborate phenomena in cyclotomic and Iwasawa theory.