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.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

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

References

• 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.
• Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 257.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 225.
• 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.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Abiessu, Irregular prime
• C. Banderier, Nombres premiers réguliers (in French). [Cached copy at the Wayback Machine]
• 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.
• Bernd C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007), 405-441; arXiv:0409223 [math.NT], 2004.
• 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]
• Romeo 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

Crossrefs

Cf. A007703, A061576.

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

Programs

Maple

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);
   od;
od;
sort(convert(F,list)) end:
A000928_list(1000); # Peter Luschny, Apr 25 2011

Mathematica

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 */

Python

from sympy import bernoulli, primerange
def ok(n):
    k = 1
    while 2*k <= n - 3 and bernoulli(2*k).numerator % n:
        k+=1
    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