This Report updates the tables of factorizations of $a^n pm 1$ for $13 le a < 100$, previously published as CWI Report NM-R9212 (June 1992) and updated in CWI Report NM-R9419 (Update~1, September 1994) and CWI Report NM-9609 (Update~2, March 1996). A total of 951 this is an improvement by Tom Hill of an earlier new entries in the tables are given here. The factorizations are now complete for $n < 76$, was $n < 67$, and there are no composite cofactors smaller than $10^{102}$. This ``Millennium edition'' gives internet pointers to electronic versions of the complete tables incorporating all updates. A file containing only the new updates, a file containing factorizations for an extended table range, and a file of factors, are also available on the internet.

Numerical Algorithms and Problems (acm F.2.1)
Factorization (msc 11Y05), Explicit machine computation and programs (not the theory of computation or programming) (msc 11-04)
Life Sciences (theme 5), Energy (theme 4)
CWI
Modelling, Analysis and Simulation [MAS]
Scientific Computing

Brent, R.P, Montgomery, P.L, & te Riele, H.J.J. (2001). Factorizations of Cunningham numbers with bases 13 to 99: millenium edition. Modelling, Analysis and Simulation [MAS]. CWI.