irregular prime

A prime p is said to be Bernoulli-irregular if and only if it divides the numerator of one of the Bernoulli numbers B_k where 0<k<p-3. An alternate definition is to say that a prime p is irregular iff it divides the class number of the algebraic number field defined by adjoining the p^th root of unity to the rationals.

Examples: 37, 59, 67, 101, 103, 131, 149, 157, ...

Among the infinite set of primes, it is known that there are infinitely many Bernoulli-irregular primes. It is not known however whether there is an infinite number of regular primes (though approximately 60% of primes whose regularity status is known are in fact regular).

Alternatively, a prime p can be said to be Euler-irregular iff it divides an Euler number E(n) (see zig number) where 1<2*n<p-1. Some Euler-irregular primes are 19 and 61.