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Irregular primes to one million. (English) Zbl 0768.11009
The authors have calculated the irregular pairs $(p,k)$ for all primes $p < 10\sp 6$. Recall that a pair $(p,k)$ is called irregular if $k$ is an even integer with $2 \leq k \leq p-3$ such that $p$ divides the numerator of the Bernoulli number $B\sb k$. Previous computations of the irregular pairs, covering the range $p < 150\ 000$ [see {\it J. W. Tanner} and {\it S. S. Wagstaff}, ibid. 48, 341--350 (1987; Zbl 0613.10012)], used algorithms requiring $O(p\sp 2)$ arithmetic operations for each prime $p$. The present authors were able to reduce this number to $O(p\log p)$. They computed Bernoulli numbers modulo $p$ basically from the formula $$\sum\sp \infty\sb{k = 0} B\sb kx\sp k/k! = (1 + x/2! + x\sp 2/3! + \dots)\sp{-1},$$ performing the power series inversion by algorithms based on the fast Fourier transform and multisectioning of power series. The maximum number of irregular pairs $(p,k)$ found in this range was 6, occurring for $p = 527377$. The authors also used their results to verify that Fermat’s “Last Theorem” and Vandiver’s conjecture are true for the primes $p < 10\sp 6$. {A subsequent work by the first two authors together with {\it R. Ernvall} and the reviewer [ibid., July 1993 issue] extends the above calculations to all primes below four million and moreover gives the ordinary cyclotomic invariants for these primes}.

11B68Bernoulli and Euler numbers and polynomials
11D41Higher degree diophantine equations
65Y20Complexity and performance of numerical algorithms
11R18Cyclotomic extensions
11Y55Calculation of integer sequences
68Q25Analysis of algorithms and problem complexity
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