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A000928 Irregular primes: primes p such that at least one of the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) is divisible by p.
(Formerly M5260 N2292)
37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593, 607, 613, 617, 619, 631, 647, 653, 659, 673, 677, 683, 691, 727, 751, 757, 761, 773, 797, 809, 811, 821, 827, 839, 877, 881, 887, 929, 953, 971, 1061 (list; graph; refs; listen; history; text; internal format)



A prime is irregular if and only if the integer Sum_{j=1..p-1} cot^(r)(j*Pi/p)*cot(j*Pi/p) is divisible by p for some even r <= p-5. (See G. Almkvist 1994.) - Peter Luschny, Jun 24 2012

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]

Johnson (1975) mentions "consecutive irregular prime pairs", meaning an irregular prime p such that, for some integer k <= 2*p-3, p divides the numerators of the Bernoulli numbers B_{2k} and B_{2k+2}. He gives the examples p = 491 (with k=168) and p = 587. No other examples are known. - N. J. A. Sloane, May 01 2021, following a suggestion from Felix Fröhlich.

An odd prime p is irregular if and only if p divides the class number of Q(zeta_p), where zeta_n = exp(2*Pi*i/n); that is, for k >= 2, p = prime(k) is irregular if and only if p divides A055513(k). For example, 37 is irregular since Q(zeta_37) has class number A055513(12) = 37. - Jianing Song, Sep 13 2022


G. Almkvist, Wilf's conjecture and a generalization, In: The Rademacher legacy to mathematics, 211-233, Contemp. Math., 166, Amer. Math. Soc., Providence, RI, 1994.

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 377, 425-430 (but there are errors in the tables).

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, see p. 244.

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

P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 257.

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 réguliers (in French).

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.

Joe P. Buhler and David Harvey, Irregular primes to 163 million

Joe P. Buhler and David Harvey, Irregular primes to 163 million, arXiv:0912.2121 [math.NT], 2009.

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.

William Hart, David Harvey and Wilson Ong, Irregular primes to two billion, Math. Comp. 86 (2017), 3031-3049. Preprint arXiv:1605.02398 [math.NT].

David Harvey, List of irregular pairs, 2017. Download size 557 MB.

Su Hu, Min-Soo Kim, Pieter Moree and Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, arXiv:1809.08431 [math.NT], 2018.

K. L. Jensen, Om talteoretiske Egenskaber ved de Bernoulliske Tal, Nyt Tidskrift für Math. Afdeling B 28 (1915), pp. 73-83.

W. Johnson, On the vanishing of the Iwasawa invariant {mu}_p for p < 8000, Math. Comp., 27 (1973), 387-396 (gives a list up to 8000 and 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.

W. Johnson, Irregular primes and cyclotomic invariants, Math. Comp. 29 (1975), 113-120.

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, Proc. Nat. Acad. Sci. USA, 40 (1954), 25-33 (but there are errors).

C. Lin and L. Zhipeng, On Bernoulli numbers and its properties, arXiv:math/0408082 [math.HO], 2004.

F. Luca, A. Pizarro-Madariaga and C. Pomerance, On the counting function of irregular primes, 2014.

Peter Luschny, The Computation of Irregular Primes. [From Peter Luschny, Apr 20 2009]

R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

Tauno Metsänkylä, Note on the distribution of irregular primes, Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, 492, 1971.

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, Irregular Prime

Eric Weisstein's World of Mathematics, Integer Sequence Primes

Index entries for sequences related to Bernoulli numbers.

Bernoulli numbers, irregularity index of primes


A000928_list := proc(len)

local ab, m, F, p, maxp; F := {};

for m from 2 by 2 to len do

p := nextprime(m+1);

ab := abs(bernoulli(m));

maxp := min(ab, len);

while p <= maxp do

if ab mod p = 0

then F := F union {p} fi;

p := nextprime(p);



sort(convert(F, list)) end:

A000928_list(1000); # Peter Luschny, Apr 25 2011


fQ[p_] := Block[{k = 1}, While[ 2k <= p-3 && Mod[ Numerator@ BernoulliB[ 2k], p] != 0, k++]; 2k <= p-3]; Select[ Prime@ Range@ 137, fQ] (* Robert G. Wilson v, Jun 25 2012 *)

Select[Prime[Range[200]], MemberQ[Mod[Numerator[BernoulliB[2*Range[(#-1)/ 2]]], #], 0]&] (* Harvey P. Dale, Mar 02 2018 *)


(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 */


from sympy import bernoulli, primerange

def ok(n):

k = 1

while 2*k <= n - 3 and bernoulli(2*k).numerator() % n:


return 2*k <= n - 3

print([n for n in primerange(2, 1101) if ok(n)]) # Indranil Ghosh, Jun 27 2017, after Robert G. Wilson v


Cf. A007703, A061576.

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

Sequence in context: A127023 A109166 A090798 * A073276 A281290 A105460

Adjacent sequences: A000925 A000926 A000927 * A000929 A000930 A000931




N. J. A. Sloane



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Last modified March 23 20:04 EDT 2023. Contains 361452 sequences. (Running on oeis4.)