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

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

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 June 15, 2013)