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An upper bound for the index of χ-irregularity

  • R. Ernvall (a1)

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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.

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References

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1.Carlitz, L.. Note on irregular primes. Proc Amer. Math. Soc., 5 (1954), 329331.
2.Ernvall, R.. Generalized Bernoulli numbers, generalized irregular primes, and class number. Ann. Univ. Turku Ser. A Math., 178 (1979), 72pp.
3.Ernvall, R.. Generalized irregular primes. Mathematika, 30 (1983), 6773.
4.Ernvall, R. and Metsänkylä, T.. Cyclotomic invariants and E-irregular primes. Math. Comp., 32 (1978), 617629. Corrigendum. Ibid., 33 (1979), 433.
5.Iwasawa, K.. Lectures on p-adic L-functions (Princeton University Press, 1972).
6.Leopoldt, H.-W.. Eine Verallgemeinerung der Bernoullischen Zahlen. Abh. Math. Sem. Univ. Hamburg, 22 (1958), 131140.
7.Skula, L.. Index of irregularity of a prime. J. Reine Angew. Math., 315 (1980), 92106.
8.Ullom, S.. Upper bounds for p-divisibility of sets of Bernoulli numbers. J. Number Theory, 12 (1980), 197200.
9.Wagstaff, S. S. Jr. The irregular primes to 125,000. Math. Comp., 32 (1978), 583591.
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