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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 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.
- 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
- 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
- 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
- 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
- L. Carlitz, "Note on irregular primes," Proc. Amer. Math. Soc., 5 (1954) 329--331. MR 15,778b
- W. Johnson, "Irregular prime divisors of the bernoulli numbers," Math. Comp., 28 (1974) 653--657. MR 50:229
- W. Johnson, "Irregular primes and cyclotomic invariants," Math. Comp., 29 (1975) 113--120. MR 51:12781
- 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.]
- C. L. Siegel, "Zu zwei Bemerkungken Kummers," Nachr. Akad. d. Wiss. Goettingen, Math. Phys. KI., II (1964) 51--62.
- J. W. Tanner and S. S. Wagstaff Jr., "New congruences for the Bernoulli numbers," Math. Comp., 48 (1987) 341--350. MR 87m:11017
- 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.]
- 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