Are there any definitions for (ir)regular primes which do **not** use class number divisibility or Bernoulli numbers? For reference, Wikipedia gives both the first definition (their primary one) and the second (*i.e.*, Kummer’s criterion, using Bernoulli numbers).

I'm looking for a third (or more) fundamentally different characterization — bonus points if it can be stated and proven in an elementary (if not necessarily simple) way. Here's a conjectured example: A prime $p$ is *irregular* if and only if there exists some even integer $2 \le n \le p-3$ such that
$$
p^2 \mid (1^n + 2^n + \dotsb + (p-1)^n + p^n).
$$

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