Author: William A. Stein
1World record calculation of Bernoulli numbers:
2
3x/(e^x-1)
4
5
6See
7
8  http://www.research.att.com/~njas/sequences/table?a=103233&fmt=4
9
10Sloane sequence A103233 = number of digits of absolute value of
11numerator of B_{10^n}.  It has been computed for n = 1,2,3,4,5,6, but
12n=7 apparently remains out of reach.
13
14  0  1
15  1  1
16  2  83
17  3  1779
18  4  27691
19  5  376772
20  6  4767554
21
22Using our algorithm the timings are
23
24  0  0
25  1  0
26  2  0
27  3  0
28  4  0.97  seconds
29  5  107   seconds
30  6  ???
31
32----------------------------
33
34http://www.bernoulli.org
35
36formula to compute B_k quickly
37
38----------------------------
39
40Quote from http://www.mathstat.dal.ca/~dilcher/bernoulli.html
41
42The Bernoulli numbers are among the most interesting and important
43number sequences in mathematics. They first appeared in the posthumous
44work "Ars Conjectandi" (1713) by Jakob Bernoulli (1654-1705) in
45connection with sums of powers of consecutive integers (see Bernoulli
46(1713) or D.E. Smith (1959)). Bernoulli numbers are particularly
47important in number theory, especially in connection with Fermat's
48last theorem (see, e.g., Ribenboim (1979)). They also appear in the
49calculus of finite differences (Nörlund (1924)), in combinatorics
50(Comtet (1970, 1974)), and in other fields.''
51
52
53----------------------------
54
55From Mazur's article: "The "Bernoulli Number" Website
56http://www.mscs.dal.ca/~dilcher/bernoulli.html offers a bibliography
57of a few thousand articles giving us a sensethat these numbers pervade
58mathematics, but to get a more vivid sense of how they do so, we will
59survey, in the lecture, the pertinence of Bernoulli numbers in just a
60few subjects."
61
62
63----------------------------
64
65From http://mathworld.wolfram.com/BernoulliNumber.html
66
67"The only known Bernoulli numbers B_n having prime numerators occur for
68n==10, 12, 14, 16, 18, 36, and 42 (Sloane's A092132), corresponding to
695, -691, 7, -3617, 43867, -26315271553053477373, and
701520097643918070802691 (Sloane's A092133), with no other primes for
71n<=55274 (E. W. Weisstein, Apr. 17, 2005)."
72
73This suggests systematic computation of Bernoulli numbers up to 55274
74was recently done.  I should be able to easily push this to 100000 and
75run PARI's isprime on the numerators using sage.ucsd.edu.
76
77
78---------------------------
79
80From http://mathworld.wolfram.com/BernoulliNumber.html
81there is also a discussion of "record" calculations.
82
83Largest one ever computed was B_{5000000}, which was done in Oct. 8,
842005, and whose numerator has 27332507 digits.
85
86----------------------------
87
88Found via
89
90http://www.mathstat.dal.ca/~dilcher/bernoulli.html
91
92http://www.emis.de/cgi-bin/jfmen/MATH/JFM/full.html?first=1&maxdocs=20&type=html&an=10.0192.01&format=complete
93
95 Table of the values of the first sixty-two numbers of Bernoulli. (English)
96 [J] Borchardt J. LXXXV. 269-272.
97 Published: 1878
98 In Crelle's J. XX.
99
100--------
101
102JFM 05.0144.02
103 Glaisher, J. W. L.
104 Tables of the first 250 Bernoulli's numbers (to nine figures) and their logarithmes (to ten figures). (English)
105 [J] Trans. of Cambridge. XII. I. 384-391.
106 Published: (1873)
107 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.
108 [ Glaisher, Prof. (Cambridge) (Ohrtmann, Dr. (Berlin)) ]
109 Subject heading: Fünfter Abschnitt. Reihen. Capitel 1. Allgemeines.
110
111-----------------
112
113SEREBRENIKOV S.Z.,
114  Tablitsy pervykh devyanosta chisel Bernulli [Tables of the first ninety Bernoulli numbers]. Zap. Akad. Nauk, Sankt Peterburg, 16 (1905), no. 10, 1-8.
115  J36.0342.02
116
117
118-----------------
119
120
121
123
124 (1) computing B_k
125     Algorithm of Cohen & Bellabas  (2004?)
126   and independently of
127     Bernd Kellner (2002-2004)
128
129   neither bothered with any proofs of correctness.
130      Kellner -- closed source C++ program and paper
131             that basically suggests he gets answer
132             then verified congruences to make
133             "morally certain".  Not totally rigorous
134
135      Bellabas -- it's a PARI function; nothing published
136             about why/how/if it really works correctly,
137             to my knowledge -- this is typical.
138
139 (2) table of B_k:
140       I scoured the web and couldn't find anything beyond
141       tables up to 20 and a couple of select huge values
142       (B_k for k up to k=2*10^5, and also claim somebody computed
143        B_k for k = 5*10^5)
144
145 (3) generalization to B_{k,chi} -- very important to
146       computing modular forms of level > 1.
147
148 (4)