Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-12-01T02:54:20.608Z Has data issue: false hasContentIssue false
  • English
  • Français

An upper bound for the index of χ-irregularity

Published online by Cambridge University Press:  26 February 2010

R. Ernvall
R. Ernvall
Affiliation:
Department of Mathematics, University of Turku, SF-20500 Turku, Finland.
Get access

Extract

In the middle of the last century, Kummer's studies on the famous Fermat conjecture led him to the question: when does a given prime p > 2 divide the class number of the p-th cyclotomic field? His conclusion was that this happens, if, and only if, p divides at least one of the Bernoulli numbers B2, B4,…, Bp_3. Such a prime is called irregular. Carlitz [1] has given the simplest proof of the fact that the number of irregular primes is infinite. However, it is not known whether there are infinitely many regular primes.

Type
Research Article
Information
Mathematika , Volume 32 , Issue 1 , June 1985 , pp. 39 - 44
Copyright
Copyright © University College London 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Carlitz, L.. Note on irregular primes. Proc Amer. Math. Soc., 5 (1954), 329–331.CrossRefGoogle Scholar
2.Ernvall, R.. Generalized Bernoulli numbers, generalized irregular primes, and class number. Ann. Univ. Turku Ser. A Math., 178 (1979), 72pp.Google Scholar
3.Ernvall, R.. Generalized irregular primes. Mathematika, 30 (1983), 67–73.CrossRefGoogle Scholar
4.Ernvall, R. and Metsänkylä, T.. Cyclotomic invariants and E-irregular primes. Math. Comp., 32 (1978), 617–629. Corrigendum. Ibid., 33 (1979), 433.Google Scholar
5.Iwasawa, K.. Lectures on p-adic L-functions (Princeton University Press, 1972).CrossRefGoogle Scholar
6.Leopoldt, H.-W.. Eine Verallgemeinerung der Bernoullischen Zahlen. Abh. Math. Sem. Univ. Hamburg, 22 (1958), 131–140.CrossRefGoogle Scholar
7.Skula, L.. Index of irregularity of a prime. J. Reine Angew. Math., 315 (1980), 92–106.Google Scholar
8.Ullom, S.. Upper bounds for p-divisibility of sets of Bernoulli numbers. J. Number Theory, 12 (1980), 197–200.CrossRefGoogle Scholar
9.Wagstaff, S. S. Jr. The irregular primes to 125,000. Math. Comp., 32 (1978), 583–591.CrossRefGoogle Scholar