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Mathematics of Computation

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Irregular primes and cyclotomic invariants


Author: Wells Johnson
Journal: Math. Comp. 29 (1975), 113-120
MSC: Primary 12A35; Secondary 10A40, 10B15
DOI: https://doi.org/10.1090/S0025-5718-1975-0376606-9
MathSciNet review: 0376606
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Abstract: The table of irregular primes less than 30000 has been computed and deposited in the UMT file. The fraction of irregular primes in this range is 0.3924, close to the heuristic prediction of $ 1 - {e^{ - 1/2}}$. Fermat's Last Theorem has been verified for all prime exponents $ p < 30000$, and the cyclotomic invariants $ {\mu _p},{\lambda _p}$, and $ {\nu _p}$ of Iwasawa have been completely determined for these primes. The computations show that for p in this range, $ {\mu _p} = 0$ and the invariants $ {\lambda _p}$ and $ {\nu _p}$ both equal the index of irregularity of p.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1975-0376606-9
Keywords: Irregular primes, Bernoulli numbers, Fermat's Last Theorem, cyclotomic fields, class numbers, $ \Gamma $-extensions, cyclotomic invariants
Article copyright: © Copyright 1975 American Mathematical Society