News (21 oct 2012):
Shi Bai and I would like to announce that we have factored the remaining C173 of the numerator of the 202th Bernoulli number (see http://homes.cerias.purdue.edu/~ssw/bernoulli/bnum for a list of known factors). We used the CADONFS program (http://cadonfs.gforge.inria.fr/). The polynomial selection took around 10 days on a 48 core 2.2GHz AMD Opteron. The sieving took around 42 days on the 48 core machine. Linear algebra took around 15 days on 72 cores on an AMD Opteron 2.3 GHz. The final square root phase consumed less than a day on 8 cores on the same machine. The C173 factored into a 76digit prime and a 97digit prime.

The Bernoulli numbers B_{n}_{ }play an important role in several topics of mathematics. These numbers can be defined by the power series
where all numbers B_{n} are zero with odd index n > 1. The evenindexed rational numbers B_{n} alternate in sign. The first values are
Sum of consecutive integer powers 
Jakob Bernoulli introduced a sequence of rational numbers, later
called Bernoulli numbers, to compute the sum of consecutive integer
powers. This formula is given by
Values of the Riemann zeta function 
The Bernoulli numbers are connected with the Riemann zeta function
on the positive real axis by Euler's formula for positive even n, also valid for n = 0:
The functional equation of ζ(s) leads to the following formula for negative integer arguments:
Structure of the denominator 
The structure of the denominator of B_{n} for positive even n is given by the Clausen  von Staudt Theorem:
Structure of the numerator 
The numerator of B_{n}/n for positive even n equals 1 only for n = 2,4,6,8,10,14; otherwise this numerator is a product of powers of irregular primes. Since B_{n}/n is a pinteger for primes p where p1 does not divide n, the structure of the numerator of B_{n}_{ }is given by
The first product is a trivial factor of B_{n} which divides n. The second product consists only of powers of irregular primes p_{v} which are not easy to determine for larger n.
Kummer congruences 
The Kummer congruences describe the most important arithmetical
properties of the Bernoulli numbers which give a modular relation
between these numbers.
The two millionth Bernoulli number 
More than 10 million digits were omitted in the middle of the numerator!
The 1.5 millionth Bernoulli number 
More than 7.4 million digits were omitted in the middle of the numerator!
The one millionth Bernoulli number 
More than 4.7 million digits were omitted in the middle of the numerator!
Program Calcbn  A multithreaded program for computing Bernoulli numbers via Riemann zeta function 
Version 
Windows (32bit) / Exe
single & multithreaded 
Linux (64bit) / Source
single & multithreaded 
2.0

calcbn32exe.zip (150 KB)

calcbn64src.zip (26 KB) 
Calcbn can be freely used without any warranty. Calcbn depends on the GMP library, so use the latest version of GMP with possible optimization for the current hardware that is being used. The source code of Calcbn is released under the terms of the GNU Public License. Timings on an Intel Core2Duo E6850 @ 3 GHz, computation of the 100000th Bernoulli number: Windows XP (32bit): 16.8s (one thread), 8.7s (two threads). Linux openSUSE 11.0 (64bit): 11.3s (one thread), 5.7s (two threads).
Factorization of numerators 
Factorization of numerators of Bernoulli numbers with index 2 to 10000. Computed prime factors are less than one million: factors10t.txt (111 KB).
Irregular pairs of higher order 
The irregular pairs of higher order describe the first appearance of higher powers of irregular prime factors of B_{n}/n. An irregular pair (p,n) of order r has the property that p^{r} divides B_{n}/n with n < (p1)p^{r1}. Note that n is always an even positive integer. For r = 1 this gives the usual definition of irregular pairs; note that the condition p divides B_{n}/n is then equal to p divides B_{n}.
There exists a criterion to check whether the sequence of irregular
pairs of higher order is unique. It has been proven for all irregular
primes below 12 million that there are only unique sequences. Writing
sequences padically these pairs of higher order provide an approximation of a uniquely existing zero of the padic zeta function associated with an irregular pair. For definition and properties see [R3]. Example:
Irregular pairs of higher order can be effectively and easily
computed using Bernoulli numbers with small indices. By this means one
can even predict the very large index of the first occurence of the
power 37^{37} as listed above. Table of irregular pairs of order 10 for irregular primes below 1000: irrpairord.txt
A conjectural structural formula for the Bernoulli numbers 
Assuming that all sequences of irregular pairs of higher order are unique, resp. each padic zeta function associated with an irregular pair (p,l) has a unique simple zero χ _{(p,l)},
one can describe the structure of divided Bernoulli numbers, resp. the
value of the Riemann zeta function at negative odd integer arguments, as
follows:
Under the proposed assumption, one can give some interpretation of
the formula above. The denominator can be described by poles (always
lying at 0) and the numerator by zeros of padic zeta functions measuring the distance to them using the padic metric induced by the standard ultrametric absolute value  _{p}.
Note that this formula is valid for all irregular primes below 12
million. For detailed statements see [R3]. Equivalently, the formula
conjecturally states for the Bernoulli numbers that
If there should exist a sequence of irregular pairs of higher
order that is not unique, – the socalled singular case –, then the
formulas given above remain valid. An additional product has to be added
for those irregular pairs which belong to the singular case. This
product can be described by trees of irregular pairs of higher order,
which is much more complicated. Since there is not known any example of
the singular case, the conjectural formulas are presented here in a
simple form.
A conjectural structural formula for the Euler numbers 
Similarly, for the Euler numbers E_{n}, n > 0 and even, one can state also a conjectural formula:
Here (p,l) are irregular pairs associated with the Euler numbers and the ξ _{(p,l)}_{ }are certain zeros of padic Lfunctions.
Connections with class numbers of imaginary quadratic fields 
Let h(d) denote the class number of the imaginary quadratic field Q(√d) of discriminant d < 4. For primes p > 3 one has the connections with Bernoulli and Euler numbers due to Carlitz [R1]:
Since h(p) < p and h(4p) < p for these cases, this shows that p cannot divide special Bernoulli and Euler numbers. That means for Bernoulli numbers that an irregular pair (p,(p+1)/2) for p ≡ 3 (mod 4) cannot exist.
Asymptotic formula 
The product of Bernoulli numbers is described by the following asymptotic formula, see [R4],
with an asymptotic constant C_{2} = 4.855096646522... which is given by
where C_{1} = 1.8210174514992... is the product over all values of the Riemann zeta function at even positive integers and A = 1.2824271291... is the GlaisherKinkelin constant:
Links & References 
 [L1] Karl Dilcher, Ilja Sh. Slavutskii: A Bibliography of Bernoulli Numbers
 [L2] Eric Weisstein: World of Mathematics, Bernoulli number
 [L3] David Harvey: Homepage. A multimodular algorithm for computing Bernoulli numbers
 [L4] S. S. Wagstaff: Factors of Bernoulli and Euler Numbers
 [L5] Chris Caldwell: Irregular primes  The Top 20
 [L6] Bernd C. Kellner: Homepage. Articles on Bernoulli numbers
 [L7] Mikko Tommila: Apfloat  High Performance Arbitrary Precision Arithmetic Package for C++ and Java
 [L8] gmplib.org: GNU Multiple Precision Arithmetic Library for C/C++
 [R1] L. Carlitz, The class number of an imaginary quadratic field, Comment. Math. Helv. 27 (1953), 338345.
 [R2] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, GTM, vol. 84, SpringerVerlag, 2nd edition, 1990.
 [R3] B. C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Comp. 76 (2007), no. 257, 405441.
 [R4] B. C. Kellner, On Asymptotic Constants Related to Products of Bernoulli Numbers and Factorials, Integers 9 (2009), 83106.
Created by Bernd C. Kellner, Göttingen, Germany.
For questions or suggestions: bk (at) bernoulli.org
Last updated: Apr. 16, 2009Ref: http://www.bernoulli.org/
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