By definition, \(p\) 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 \(p\nmid h(K)\). Applying theorem 2.1, the negation of \(p\mid h(K)\) is exactly the assertion that there is no \(k\) in the range \(1\le k\), \(2k\le p-3\) for which \(p\mid \operatorname {num}(B_{2k})\). This is the displayed universal nondivisibility condition.