regular prime
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000: 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 conjectureHeuristically 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:

BCEM1993
J. Buhler, R. Crandall, R. Ernvall and T. Metsänkylä, "Irregular primes and cyclotomic invariants to four million," Math. Comp., 61:203 (1993) 151--153.  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., 31:1--2 (2001) 89--96.  MR 2001m:11220
BCS1992
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
BH2011
Buhler, J. P. and Harvey, D., "Irregular primes to 163 million," Math. Comp., 80:276 (2011) 2435--2444.  (http://dx.doi.org/10.1090/S0025-5718-2011-02461-0) MR 2813369
Carlitz1954
L. Carlitz, "Note on irregular primes," Proc. Amer. Math. Soc., 5 (1954) 329--331.  MR 15,778b
Johnson1974
W. Johnson, "Irregular prime divisors of the bernoulli numbers," Math. Comp., 28 (1974) 653--657.  MR 50:229
Johnson1975
W. Johnson, "Irregular primes and cyclotomic invariants," Math. Comp., 29 (1975) 113--120.  MR 51:12781
Ribenboim95
P. Ribenboim, The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995.  pp. xxiv+541, ISBN 0-387-94457-5. 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 seventy-five pages.]
Siegel1964
C. L. Siegel, "Zu zwei Bemerkungken Kummers," Nachr. Akad. d. Wiss. Goettingen, Math. Phys. KI., II (1964) 51--62.
TW1987
J. W. Tanner and S. S. Wagstaff Jr., "New congruences for the Bernoulli numbers," Math. Comp., 48 (1987) 341--350.  MR 87m:11017
Wagstaff78
Wagstaff, Jr., S. S., "The irregular primes to 125,000," Math. Comp., 32 (1978) 583-591.  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, Springer-Verlag, New York, NY, 1982.  pp. xi+389, ISBN 0-387-90622-3. (There is a later edition).  MR 85g:11001



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