The mathematician Kummer called a prime regular if it does not
divide the class number of the algebraic number field defined by adjoining
a pth root of unity to the rationals. Since this may mean
little to most of the readers of this glossary, let us quickly add that
Kummer was able to show p was regular if (and only if) it does
not divide the numerator of any of the Bernoulli numbers Bk for k=2, 4, 6, ..., p-3. For example, 691 divides
the numerator of B12, so 691 is not regular (we say it is irregular).
Kummer was interested in these numbers because he could show that if
n was divisible by a regular prime, then Fermat's Last Theorem
was true for that n. Algebraic number theory and Kummer's
ideal theory are just two more of the many fields which this one
problem gave a great boost!
The first few irregular primes (those which are not regular)
are 37, 59, 67, 101, 103, 131, 149 and 157 (which is the first to
divide two). It is relatively easy to show that there are
infinitely many irregular primes, but the infinitude of regular
primes is still just a conjecture. Heuristically we estimate
that e-1/2 (about 60.65%) of the primes are regular.
To check this estimate Wagstaff found all of the regular primes below
125,000 and found that they compose 60.75% of those primes.
The irregularity index of a prime p is the number of
times that p divides the Bernoulli numbers B(2n) for
1 < 2n < p-1. The irregularity index of
157 is 2 because 157 divides B(62) and B(110). Regular primes
have an irregularity index of zero.
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