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

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

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

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

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

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

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

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

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.