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Author: William A. Stein
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World record calculation of Bernoulli numbers:
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x/(e^x-1)
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See
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http://www.research.att.com/~njas/sequences/table?a=103233&fmt=4
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Sloane sequence A103233 = number of digits of absolute value of
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numerator of B_{10^n}. It has been computed for n = 1,2,3,4,5,6, but
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n=7 apparently remains out of reach.
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0 1
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1 1
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2 83
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3 1779
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4 27691
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5 376772
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6 4767554
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Using our algorithm the timings are
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0 0
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1 0
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2 0
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3 0
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4 0.97 seconds
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5 107 seconds
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6 ???
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http://www.bernoulli.org
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formula to compute B_k quickly
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Quote from http://www.mathstat.dal.ca/~dilcher/bernoulli.html
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``The Bernoulli numbers are among the most interesting and important
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number sequences in mathematics. They first appeared in the posthumous
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work "Ars Conjectandi" (1713) by Jakob Bernoulli (1654-1705) in
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connection with sums of powers of consecutive integers (see Bernoulli
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(1713) or D.E. Smith (1959)). Bernoulli numbers are particularly
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important in number theory, especially in connection with Fermat's
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last theorem (see, e.g., Ribenboim (1979)). They also appear in the
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calculus of finite differences (Nörlund (1924)), in combinatorics
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(Comtet (1970, 1974)), and in other fields.''
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From Mazur's article: "The "Bernoulli Number" Website
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http://www.mscs.dal.ca/~dilcher/bernoulli.html offers a bibliography
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of a few thousand articles giving us a sensethat these numbers pervade
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mathematics, but to get a more vivid sense of how they do so, we will
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survey, in the lecture, the pertinence of Bernoulli numbers in just a
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few subjects."
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From http://mathworld.wolfram.com/BernoulliNumber.html
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"The only known Bernoulli numbers B_n having prime numerators occur for
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n==10, 12, 14, 16, 18, 36, and 42 (Sloane's A092132), corresponding to
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5, -691, 7, -3617, 43867, -26315271553053477373, and
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1520097643918070802691 (Sloane's A092133), with no other primes for
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n<=55274 (E. W. Weisstein, Apr. 17, 2005)."
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This suggests systematic computation of Bernoulli numbers up to 55274
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was recently done. I should be able to easily push this to 100000 and
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run PARI's isprime on the numerators using sage.ucsd.edu.
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From http://mathworld.wolfram.com/BernoulliNumber.html
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there is also a discussion of "record" calculations.
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Largest one ever computed was B_{5000000}, which was done in Oct. 8,
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2005, and whose numerator has 27332507 digits.
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Found via
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http://www.mathstat.dal.ca/~dilcher/bernoulli.html
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http://www.emis.de/cgi-bin/jfmen/MATH/JFM/full.html?first=1&maxdocs=20&type=html&an=10.0192.01&format=complete
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Adams, J. C.
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Table of the values of the first sixty-two numbers of Bernoulli. (English)
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[J] Borchardt J. LXXXV. 269-272.
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Published: 1878
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In Crelle's J. XX.
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--------
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JFM 05.0144.02
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Glaisher, J. W. L.
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Tables of the first 250 Bernoulli's numbers (to nine figures) and their logarithmes (to ten figures). (English)
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[J] Trans. of Cambridge. XII. I. 384-391.
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Published: (1873)
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Enthält zwei Tafeln: die erste für die Logarithmen der 250 ersten Bernoulli'schen Zahlen zu zehn Stellen, die alle, mit Ausnahme der sieben ersten, berechnet sind aus der Formel: $$B_n = \frac{2(1\cdot 2\cdots n)}{(2\pi )^{2n}} \left( 1+ \frac {1}{2^{2n}}+ \frac {1}{3^{2n}} + \cdots \right) .$$ Die zweite Tafel enthält die ersten neun Stellen der betreffenden Bernoulli'schen Zahlen, hergeleitet aus der ersten Tafel, (ausgenommen die ersten achtzehn, welche aus den genauen von Ohm, Crelle J. XII, gegebenen Werthen hergeleitet sind). Der Verfasser bemerkt, dass die kleine Tafel in Grunert's Supplement zu Klügel's Wörterbuch sehr ungenau ist, indem sieben von den achtzehn Resultaten mit Fehlern behaftet sind.
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[ Glaisher, Prof. (Cambridge) (Ohrtmann, Dr. (Berlin)) ]
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Subject heading: Fünfter Abschnitt. Reihen. Capitel 1. Allgemeines.
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SEREBRENIKOV S.Z.,
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[1] Tablitsy pervykh devyanosta chisel Bernulli [Tables of the first ninety Bernoulli numbers]. Zap. Akad. Nauk, Sankt Peterburg, 16 (1905), no. 10, 1-8.
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J36.0342.02
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This talk is about
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(1) computing B_k
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Algorithm of Cohen & Bellabas (2004?)
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and independently of
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Bernd Kellner (2002-2004)
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neither bothered with any proofs of correctness.
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Kellner -- closed source C++ program and paper
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that basically suggests he gets answer
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then verified congruences to make
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"morally certain". Not totally rigorous
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Bellabas -- it's a PARI function; nothing published
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about why/how/if it really works correctly,
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to my knowledge -- this is typical.
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(2) table of B_k:
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I scoured the web and couldn't find anything beyond
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tables up to 20 and a couple of select huge values
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(B_k for k up to k=2*10^5, and also claim somebody computed
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B_k for k = 5*10^5)
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(3) generalization to B_{k,chi} -- very important to
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computing modular forms of level > 1.
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(4)