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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, ..., p3. For example, 691 divides
the numerator of B_{12}, 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 < p1. The irregularity index of
157 is 2 because 157 divides B(62) and B(110). Regular primes
have an irregularity index of zero.
References:
 BCEM1993
 J. Buhler, R. Crandall, R. Ernvall and T. Metsänkylä, "Irregular primes and cyclotomic invariants to four million," Math. Comp., 61:203 (1993) 151153. MR 93k:11014
 BCEMS2000
 J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla and M. Shokrollahi, "Irregular primes and cyclotomic invariants to 12 million," J. Symbolic Comput., 31:12 (2001) 8996. MR 2001m:11220
 BCS1992
 J. P. Buhler, R. E. Crandall and R. W. Sompolski, "Irregular primes to one million," Math. Comp., 59:200 (1992) 717722. MR 93a:11106
 BH2011
 Buhler, J. P. and Harvey, D., "Irregular primes to 163 million," Math. Comp., 80:276 (2011) 24352444. (http://dx.doi.org/10.1090/S002557182011024610) MR 2813369
 Carlitz1954
 L. Carlitz, "Note on irregular primes," Proc. Amer. Math. Soc., 5 (1954) 329331. MR 15,778b
 Johnson1974
 W. Johnson, "Irregular prime divisors of the bernoulli numbers," Math. Comp., 28 (1974) 653657. MR 50:229
 Johnson1975
 W. Johnson, "Irregular primes and cyclotomic invariants," Math. Comp., 29 (1975) 113120. MR 51:12781
 Ribenboim95
 P. Ribenboim, The new book of prime number records, 3rd edition, SpringerVerlag, New York, NY, 1995. pp. xxiv+541, ISBN 0387944575. MR 96k:11112 [An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventyfive pages.]
 Siegel1964
 C. L. Siegel, "Zu zwei Bemerkungken Kummers," Nachr. Akad. d. Wiss. Goettingen, Math. Phys. KI., II (1964) 5162.
 TW1987
 J. W. Tanner and S. S. Wagstaff Jr., "New congruences for the Bernoulli numbers," Math. Comp., 48 (1987) 341350. MR 87m:11017
 Wagstaff78
 Wagstaff, Jr., S. S., "The irregular primes to 125,000," Math. Comp., 32 (1978) 583591. MR 58:10711 [Kummer was able to show that FLT was true for the regular primes.]
 Washington82
 L. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics Vol, 83, SpringerVerlag, New York, NY, 1982. pp. xi+389, ISBN 0387906223. (There is a later edition). MR 85g:11001
