A007703 Regular primes.

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281, 313, 317, 331, 337, 349, 359, 367, 373, 383, 397, 419, 431

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• nice

Comments

A prime p is regular if and only if the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) are not divisible by p.

References

• Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430.
• H. M. Edwards, Fermat's Last Theorem, Springer, 1977.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 223-227.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• C. K. Caldwell, The Prime Glossary, Regular prime.
• K. Conrad, Fermat's Last Theorem For Regular Primes.
• Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
• F. Luca, A. Pizarro-Madariaga, and C. Pomerance, On the counting function of irregular primes, 2014.
• O. A. Ivanova, Regular prime number.
• D. Jao, PlanetMath.org, Regular prime.
• A. L. Robledo, PlanetMath.org, Examples of regular primes.
• Eric Weisstein's World of Mathematics, Regular Prime.
• Bernoulli numbers, irregularity index of primes

Crossrefs

Cf. A000928 (irregular primes) and A061576 for further references.

Programs

Mathematica

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, 80}]; s (* Robert G. Wilson v Sep 20 2006 *)

PARI

is(p)=forstep(k=2,p-3,2,if(numerator(bernfrac(k))%p==0, return(0))); isprime(p) \\ Charles R Greathouse IV, Feb 25 2014

Python

from sympy import prime, isprime, bernoulli
def ok(n):
    for k in range(2, n - 2, 2):
        if bernoulli(k).as_numer_denom()[0] % n == 0:
            return 0
    return isprime(n)
[n for n in range(3, 501) if ok(n)] # Indranil Ghosh, Jun 27 2017, after Charles R Greathouse IV

Extensions

Corrected by Gerard Schildberger, Jun 01 2004