Search: id:A000928 Results 1-1 of 1 results found. %I A000928 M5260 N2292 %S A000928 37,59,67,101,103,131,149,157,233,257,263,271,283,293,307,311,347, %T A000928 353,379,389,401,409,421,433,461,463,467,491,523,541,547,557,577,587,593, %U A000928 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 %N A000928 Irregular primes: 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. %C A000928 Jensen proved in 1915 that there are infinitely many irregular primes. It is not known if there are infinitely many regular primes. %C A000928 "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] %D A000928 Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966, pp. 425-430 (but there are errors). %D A000928 R. E. Crandall, Mathematica for the Sciences, Addison-Wesley Publishing Co., Redwood City, CA, 1991, pp. 248-255. %D A000928 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 59, p. 21, Ellipses, Paris 2008. %D A000928 H. M. Edwards, Fermat's Last Theorem, Springer, 1977. %D A000928 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). %D A000928 W. Johnson, Irregular prime divisors of the Bernoulli numbers, Math. Comp. 28 (1974), 653-657. %D A000928 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). %D A000928 J. Neukirch, Algebraic Number Theory, Springer, 1999, p. 38. %D A000928 P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 137. %D A000928 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A000928 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A000928 L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 350. %H A000928 T. D. Noe, Table of n, a(n) for n = 1..10000 %H A000928 Abiessu, Irregular prime %H A000928 C. Banderier, Nombres premiers reguliers %H A000928 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. %H A000928 J. P. Buhler, R. E. Crandall and R. W. Sompolski, Irregular primes to one million, Math. Comp. 59 no 200 (1992) 717-722. %H A000928 C. K. Caldwell, The Prime Glossary, Regular prime %H A000928 C. K. Caldwell, the top twenty, Irregular Primes %H A000928 V. A. Demyanenko, Irregular prime number %H A000928 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 %H A000928 B. C. Kellner, On Irregular Prime Power Divisors of the Bernoulli Numbers, Math. Comp. 75 (2006) PII S0025-5718(06)01887-4 %H A000928 D. H. Lehmer et al., An Application Of High-Speed Computing To Fermat's Last Theorem %H A000928 C. Lin and L. Zhipeng, On Bernoulli numbers and its properties %H A000928 Peter Luschny, The Computation of Irregular Primes. [From Peter Luschny (peter(AT)luschny.de), Apr 20 2009] %H A000928 H. S. Vandiver, Note On The Divisors Of The Numerators Of Bernoulli's Numbers %H A000928 H. S. Vandiver, Summary Of Results And Proofs Concerning Fermat's Last Theorem %H A000928 H. S. Vandiver, Summary Of Results And Proofs Concerning Fermat's Last Theorem %H A000928 H. S. Vandiver, Summary Of Results And Proofs Concerning Fermat's Last Theorem %H A000928 H. S. Vandiver, Summary Of Results And Proofs On Fermat's Last Theorem %H A000928 H. S. Vandiver, Summary Of Results And Proofs On Fermat's Last Theorem %H A000928 H. S. Vandiver, Summary Of Results And Proofs On Fermat's Last Theorem %H A000928 H. S. Vandiver, Examination Of Methods Of Attack On The Second Case Of Fermat's Last Theorem %H A000928 S. S. Wagstaff, Jr, The Irregular Primes to 125000, Math. Comp. 32 no 142 (1978) 583-592 %H A000928 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000928 Eric Weisstein's World of Mathematics, Integer Sequence Primes %H A000928 Index entries for sequences related to Bernoulli numbers. %H A000928 Bernoulli numbers, irregularity index of primes %t A000928 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} ] %t A000928 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 *) %o A000928 (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 %Y A000928 Cf. A007703, A061576. %Y A000928 Cf. A091887 (irregularity index of the n-th irregular prime). %Y A000928 Sequence in context: A127023 A109166 A090798 this_sequence A073276 A105460 A141851 %Y A000928 Adjacent sequences: A000925 A000926 A000927 this_sequence A000929 A000930 A000931 %K A000928 nonn,nice,easy %O A000928 1,1 %A A000928 N. J. A. Sloane (njas(AT)research.att.com). %E A000928 Johnson (1973) gives a list up to 8000. Search completed in 0.002 seconds