This site contains the full version of a paper, "Prime divisors of
the Bernoulli and Euler numbers," whose abbreviated version was
published in the Proceedings of the Millennial Conference on
Number Theory, held at the University of Illinois, Urbana, Illinois,
May 21--26, 2000.
The paper appears on pages 357--374 of volume III of Number Theory
for the Millennium, A K Peters, 2002.

It also contains full versions of tables from that paper. They
give the known prime factors of the Bernoulli numerators with
subscript up to 300, and those of the Euler numbers with subscript
up to 200.

.dvi file for paper on Bernoulli and Euler numbers

.ps file for paper on Bernoulli and Euler numbers

.pdf file for paper on Bernoulli and Euler numbers

Please send me new prime factors of the Bernoulli and Euler numbers
in the following tables, but not factors of Bernoulli and Euler numbers
with larger subscripts.

text file with factors of Bernoulli numbers
N228 was factored by NFS@Home using GNFS.

text file with factorization of the Bernoulli number B200.

text file report on the factorization of the Bernoulli number B240.

text file report on the factorization of the Bernoulli number B188.

text file with factors of Euler numbers
E148, E162 and E192 were factored by NFS@Home using GNFS.

Click
here for more information
about the Bernoulli numbers.

We have proved primality of all primes in these two tables.
We assume that anyone can prove that a prime of up to 12 digits is prime.
Old-fashioned primality proofs based on converses to Fermat's theorem
have been given for 292 of the 315 primes with at least 13 digits
in the two tables.

These 292 proofs are presented
in this file.
The notation is the same as that used in
the Cunningham book.

The other 23 large primes are
shown in this file.

We have used Francois Morain's ECPP program to prove that they are prime.
The ECPP certificates of their primality are
shown in this file (640K bytes).

The full output of Morain's ECPP program for these proofs is
shown in this file (1685K bytes).

New polynomial-time primality proving algorithm.

This text file
lists the remaining composite Bernoulli and Euler numbers,
as well as a
few Bell numbers.
If you factor any of these numbers, please send me the factor
and tell me which number it divides.

Send e-mail to Sam Wagstaff

#### (This page last modified October 27, 2023)