zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Irregular primes and cyclotomic invariants to 12 million. (English) Zbl 1001.11061

Let p be an odd prime. A pair (p,2t) (1t(p-3)/2, t) is said to be irregular for p if p divides the Bernoulli number B 2t . The number i(p) of irregular pairs of p is called the index of irregularity of p. The prime p is regular in case i(p)=0 and if i(p)1, p is irregular.

In 1857 E. E. Kummer had found out that the primes 37, 59, and 67 are irregular, and in 1879 he made the computation of irregular primes up to 163 (probably by hand). Since then, many mathematicians have continued these computations using better computational tools (calculators, computers) using increasingly better and more effective methods.

The presented results on computations of i(p) for p up to 12 million use two different algorithms. The first one is based on the power series method combined with enhanced multisectioning and convolution algorithms used in the last tables by the first four authors [Math. Comput. 61, 151-153 (1993; Zbl 0789.11020)]. The second method is a novel approach originated in the study of Stickelberger codes in [M. A. Shokrollahi, Des. Codes Cryptography 9, 203-213 (1996; Zbl 0866.94022)].

In this paper the indices of irregularity are given for primes up to 12 million. The index i(p) for these primes equal 0 to 7. Three new irregular primes with this index equal to 7 were found to one known prime with this property. Further, the Kummer-Vandiver conjecture was verified, that is the class number of the field (cos(2π/p)) is prime to p. No counterexample was found. At the conclusion the cyclotomic invariants were calculated.


MSC:
11Y40Algebraic number theory computations
11-04Machine computation, programs (number theory)