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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Irregular prime divisors of the Bernoulli numbers


Author: Wells Johnson
Journal: Math. Comp. 28 (1974), 653-657
MSC: Primary 10A40; Secondary 12A35, 12A50
DOI: https://doi.org/10.1090/S0025-5718-1974-0347727-0
MathSciNet review: 0347727
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Abstract: If p is an irregular prime, $p < 8000$, then the indices 2n for which the Bernoulli quotients ${B_{2n}}/2n$ are divisible by ${p^2}$ are completely characterized. In particular, it is always true that $2n > p$ and that ${B_{2n}}/2n\;\nequiv ({B_{2n + p - 1}}/2n + p - 1)\pmod {p^2}$ if (p,2n) is an irregular pair. As a result, we obtain another verification that the cyclotomic invariants ${\mu _p}$ of Iwasawa all vanish for primes $p < 8000$.


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Keywords: Bernoulli numbers, irregular primes, cyclotomic invariants
Article copyright: © Copyright 1974 American Mathematical Society