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<p>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, in particular, a dynamical characterisation of the Bernoulli regular primes. Algebraic realizability of the Bernoulli denominators is shown at every prime, giving a different perspective on the arithmetic structure of this sequence. We show that the sequence of Euler numbers cannot be realized on a nilpotent group, marking a fundamental difference between the nature of the Bernoulli and Euler sequences.</p>

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Local structure of classical sequences, regular primes, and dynamics

  • 期刊论文

  • 发表日期：

    2026年5月20日

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  • Sawian Jaidee,
  • Patrick Moss,
  • Thomas Ward

摘要

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, in particular, a dynamical characterisation of the Bernoulli regular primes. Algebraic realizability of the Bernoulli denominators is shown at every prime, giving a different perspective on the arithmetic structure of this sequence. We show that the sequence of Euler numbers cannot be realized on a nilpotent group, marking a fundamental difference between the nature of the Bernoulli and Euler sequences.