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 $$ p^2 \mid (1^n + 2^n + \dotsb + (p-1)^n + p^n). $$