By definition 1.1, \(p\) is regular if and only if \(p\) is coprime to the class number of the \(p\)th cyclotomic field; since \(p\) is prime, this is equivalent to \(p\nmid h(K)\). By theorem 2.1, the negation of \(p\mid h(K)\) is exactly the assertion that no \(k\) in the range \(1\le k\), \(2k\le p-3\) satisfies \(p\mid \operatorname {num}(B_{2k})\), which is the displayed universal nondivisibility.