# 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

*p*th 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 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(2*n*) for
1 < 2*n* < *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. ErnvallandT. 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. MetsankylaandM. 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. CrandallandR. W. Sompolski, "Irregular primes to one million,"Math. Comp.,59:200 (1992) 717--722.MR 93a:11106- BH2011
Buhler, J. P.andHarvey, 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, 1995. New York, NY, 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. TannerandS. 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