The irregular primes to 125000

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Abstract
We have determined the irregular primes below 125000 and tabulated their distribution. Two primes of index five of irregularity were found, namely 78233 and 94693. Fermat’s Last Theorem has been verified for all exponents up to 125000. We computed the cyclotomic invariants μ p {\mu _p} , λ p {\lambda _p} , ν p {\nu _p} , and found that μ p = 0 {\mu _p} = 0 for all p > 125000 p > 125000 . The complete factorizations of the numerators of the Bernoulli numbers B 2 k {B_{2k}} for 2 k ⩽ 60 2k \leqslant 60 and of the Euler numbers E 2 k {E_{2k}} for 2 k ⩽ 42 2k \leqslant 42 are given.

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