1 From Bernoulli numerators to irregular primes
Kummer’s criterion turns irregularity of an odd prime into a finite Bernoulli-numerator check. We record both directions in the form used by the infinitude argument, together with the finite-escape principle that drives the final contradiction.
Let \(p\) be an odd prime. If there is a \(k\) with \(1\le k\), \(2k\le p-3\) and \(p\mid \operatorname {num}(B_{2k})\), then \(p\) is not regular.
Immediate from theorem 3.1.
Conversely, if an odd prime \(p\) is not regular, then there is a \(k\) with \(1\le k\), \(2k\le p-3\) and \(p\mid \operatorname {num}(B_{2k})\).
Immediate from theorem 3.1.
Let \(P\) be a predicate on the natural numbers. If no finite set contains every \(n\) with \(P(n)\), then \(\{ n \mid P(n)\} \) is infinite.
If the set were finite, it would itself be a finite covering set.