1,1

Jensen proved in 1915 that there are infinitely many irregular primes. It is not known if there are infinitely many regular primes.

"The pioneering mathematician Kummer, over the period 1847-1850, used his profound theory of cyclotomic fields to establish a certain class of primes called 'regular' primes. ... It is known that there exist an infinity of irregular primes; in fact it is a plausible conjecture that only an asymptotic fraction 1/Sqrt(e) ~ 0.6 of all primes are regular" [Ribenboim]

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430 (but there are errors).

R. E. Crandall, Mathematica for the Sciences, Addison-Wesley Publishing Co., Redwood City, CA, 1991, pp. 248-255.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 59, p. 21, Ellipses, Paris 2008.

H. M. Edwards, Fermat's Last Theorem, Springer, 1977.

W. Johnson, On the vanishing of the Iwasawa invariant {mu}_p for p < 8000, Math. Comp., 27 (1973), 387-396 (points out that 1381, 1597, 1663, 1877 were omitted from earlier lists).

W. Johnson, Irregular prime divisors of the Bernoulli numbers, Math. Comp. 28 (1974), 653-657.

D. H. Lehmer et al., An application of high-speed computing to Fermat's last theorem, Proc. Nat. Acad. Sci. USA, 40 (1954), 25-33 (but there are errors).

J. Neukirch, Algebraic Number Theory, Springer, 1999, p. 38.

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 350.

T. D. Noe, Table of n, a(n) for n = 1..10000

Abiessu, Irregular prime

C. Banderier, Nombres premiers reguliers

J. P. Buhler, R. E. Crandall, R. Ernvall et al., Irregular primes and cyclotomic invariants to 12 Million,J. Symbolic Computation 31 (2001) 89-96.

J. P. Buhler, R. E. Crandall and R. W. Sompolski, Irregular primes to one million, Math. Comp. 59 no 200 (1992) 717-722.

C. K. Caldwell, The Prime Glossary, Regular prime

C. K. Caldwell, the top twenty, Irregular Primes

V. A. Demyanenko, Irregular prime number

G. Everest, A. J. van der Poorten, Y. Puri and T. Ward, Integer Sequences and Periodic Points, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.3

B. C. Kellner, On Irregular Prime Power Divisors of the Bernoulli Numbers, Math. Comp. 75 (2006) PII S0025-5718(06)01887-4

D. H. Lehmer et al., An Application Of High-Speed Computing To Fermat's Last Theorem

C. Lin and L. Zhipeng, On Bernoulli numbers and its properties

Peter Luschny, The Computation of Irregular Primes. [From Peter Luschny (peter(AT)luschny.de), Apr 20 2009]

H. S. Vandiver, Note On The Divisors Of The Numerators Of Bernoulli's Numbers

H. S. Vandiver, Summary Of Results And Proofs Concerning Fermat's Last Theorem

H. S. Vandiver, Summary Of Results And Proofs Concerning Fermat's Last Theorem

H. S. Vandiver, Summary Of Results And Proofs Concerning Fermat's Last Theorem

H. S. Vandiver, Summary Of Results And Proofs On Fermat's Last Theorem

H. S. Vandiver, Summary Of Results And Proofs On Fermat's Last Theorem

H. S. Vandiver, Summary Of Results And Proofs On Fermat's Last Theorem

H. S. Vandiver, Examination Of Methods Of Attack On The Second Case Of Fermat's Last Theorem

S. S. Wagstaff, Jr, The Irregular Primes to 125000, Math. Comp. 32 no 142 (1978) 583-592

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Integer Sequence Primes

Index entries for sequences related to Bernoulli numbers.

Bernoulli numbers, irregularity index of primes

Do[ p = Prime[ n ]; k = 1; While[ 2*k <= p - 3 && Mod[ Numerator[ BernoulliB[ 2*k ] ], p ] != 0, k++ ]; If[ 2*k != p - 1, Print[ p ] ], { n, 3, 200} ]

s = {}; Do[p = Prime@n; k = 1; While[2k <= p - 3 && Mod[Numerator@BernoulliB[2k], p] != 0, k++ ]; If[2k <= p - 3, AppendTo[s, p]], {n, 2, 137}]; s (* Robert G. Wilson v *)

(PARI) a(n)=local(p); if(n<1, 0, p=a(n-1)+(n==1); while(p=nextprime(p+2), forstep(i=2, p-3, 2, if(numerator(bernfrac(i))%p==0, break(2)))); p) - Michael Somos Feb 04 2004

Cf. A007703, A061576.

Cf. A091887 (irregularity index of the n-th irregular prime).

Sequence in context: A127023 A109166 A090798 this_sequence A073276 A105460 A141851

Adjacent sequences: A000925 A000926 A000927 this_sequence A000929 A000930 A000931

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

Johnson (1973) gives a list up to 8000.