Local structure of classical sequences, regular sequences, and dynamics

We introduce the notions of local realizability at a prime and algebraic realizability of an integer sequence. After discussing this notion in general we consider it for the Euler numbers, the Bernoulli denominators, and the Bernoulli numerators. This gives, for example, a dynamical characterization of the Bernoulli regular primes. Algebraic realizability of the Bernoulli denominators is shown at every prime, giving a different perspective on the great diversity of congruences satisfied by this sequence. We show that the sequence of Euler numbers cannot be realized on a nilpotent group, which may explain why it is less hospitable to congruence hunting.

💡 Research Summary

The paper introduces two complementary notions for studying integer sequences: local realizability at a prime and algebraic realizability. A sequence a = (aₙ)ₙ≥1 is globally realizable if there exists a dynamical system (X,T) such that aₙ equals the number of n‑periodic points of T. It is locally realizable at a prime q if the sequence of q‑adic valuations ⌊aₙ⌋_q is itself realizable. Algebraic realizability means that a can be obtained as the fixed‑point count of an endomorphism (or automorphism) of a group X; the group may be required to belong to a particular class (abelian, nilpotent, etc.).

Two standard ways to verify realizability are discussed. The “serendipity” approach relies on an explicit map T, while the “brute‑force” method checks the Dold congruences and the sign condition: for every n, Σ_{d|n} μ(d)·a_{n/d} must be non‑negative and divisible by n. The authors point out that Arias de Reyna’s theorem dramatically simplifies the Dold check: it suffices to verify that for each prime p with (n,p)=1 and each m≥1, a_{np^m} ≡ a_{np^{m-1}} (mod p^m).

A key structural result (Lemma 1) shows that any algebraically realizable sequence can be realized by an automorphism of a countable locally finite group, and if the sequence is bounded it can be realized on a finite group of the same size. This bridges the gap between abstract dynamical realizability and concrete group actions.

The paper then focuses on nilpotent realizability. A sequence is called a p‑sequence if each term is a pure power of p (i.e., aₙ = p^{ord_p(aₙ)}). The authors prove Theorem 4: a sequence is nilpotently realizable if and only if it is locally nilpotently realizable at every prime. The proof proceeds by showing that an algebraic realization by a group with a unique Sylow p‑subgroup forces the p‑part of the sequence to be realized on that Sylow subgroup (Lemma 9), and that locally finite nilpotent groups have unique Sylow subgroups composed of all p‑power elements (Lemma 10). The forward direction uses Lemmas 5–6 to assemble local p‑realizations into a global nilpotent group via a Cartesian product of the local groups.

With this framework the authors examine three classical families:

1. Euler numbers (absolute values of the Euler zigzag numbers). Although the Euler sequence is globally realizable (e.g., by a suitable permutation), the authors prove it cannot be realized by any nilpotent group. This explains why Euler numbers exhibit fewer systematic congruences compared with Bernoulli numbers.

2. Bernoulli denominators. For each prime p, the p‑part of the denominator of the Bernoulli number Bₙ is shown to be algebraically realizable. Consequently, every prime is “regular” in the dynamical sense: the p‑part can be obtained from a nilpotent group. This provides a new dynamical characterization of the classical notion of Bernoulli regular primes.

3. Bernoulli numerators. The situation is more delicate; only certain primes admit an algebraic realization of the numerator p‑part. This mirrors the known irregular primes in Kummer’s theory.

Concrete examples illustrate the theory. Example 3 uses the outer automorphism of the dihedral group D₈ to realize the sequence (4,4,4,8,…), showing that nilpotent realizability strictly exceeds abelian realizability. Example 11 studies the automorphism of the 3‑torus T³ defined by the matrix