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Generalized irregular primes

Published online by Cambridge University Press:  26 February 2010

Reijo Ernvall
Reijo Ernvall
Affiliation:
Department of Mathematics, University of Turku, SF-20500 Turku, Finland
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Extract

A prime p > 2 is called irregular, if it divides the numerator of at least one of the Bernoulli numbers B2, B4, …, Bp – 3 (in the even suffix notation). The study of irregular primes has its origin in the famous theorem of Kummer which states that p divides the class number of the p-th cyclotomic field, if, and only if, p is irregular. Carlitz [1] has given the simplest proof of the fact that the number of irregular primes is infinite.

Type
Research Article
Information
Mathematika , Volume 30 , Issue 1 , June 1983 , pp. 67 - 73
Copyright
Copyright © University College London 1983

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References

1.Carlitz, L.. Note on irregular primes. Proc. Amer. Math. Soc, 5 (1954), 329–331.Google Scholar
2.Ernvall, R.. Generalized Bernoulli numbers, generalized irregular primes, and class number. Ann. Univ. Turku Ser. A Math., 178 (1979), 72 pp.Google Scholar
3.Iwasawa, K.. Lectures on p-adic L-functions (Princeton University Press, 1972).CrossRefGoogle Scholar
4.Leopoldt, H.-W.. Eine Verallgemeinerung der Bernoullischen Zahlen. Abh. Math. Sem. Univ. Hamburg, 22 (1958), 131–140.Google Scholar