regular prime

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 B_k 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.

References:

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J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla and M. Shokrollahi, "Irregular primes and cyclotomic invariants to 12 million," J. Symbolic Comput., 31:1--2 (2001) 89--96.  MR 2001m:11220

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J. P. Buhler, R. E. Crandall and R. W. Sompolski, "Irregular primes to one million," Math. Comp., 59:200 (1992) 717--722.  MR 93a:11106

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