June 23, 1986


We got some new readers from the publicity in Science magazine. I hope
they know that this "newsletter" updates the book Factorizations of
b^n +- 1 by J. Brillhart et al., 1983, AMS Contemporary Math. Series,
v. 22. Pages of "New Factors," like the enclosed Page 45, are mailed
whenever one is filled (about 60 lines), which has been approximately
monthly during the past year. Once a year I send you an "official"
Update like the one stuck in the back of the book. The next one, Update
# 4, will appear in a few weeks and will include all new factors
through Page 45 and maybe a few more.

There were many exciting factorizations on Page 45. Silverman factored
the "Most Wanted" numbers 2,293+ and 6,106+. He did the "More Wanted"
numbers 7,109-, 3,158+, 3,193-, 11,79-, 11,83+, 2,538M and 5,137+.
Montgomery knocked off the "More Wanted" 12,79+ and te Riele split the
"More Wanted" 10,108+ and 2,542L. Silverman factored 3,197-, the first
hole in its table.  Montgomery and Silverman factored 12,113- and
5,143-, which were the second holes in their tables.  Silverman filled
several third and fourth holes.

te Riele's factorizations of 10,108+ and 2,542L would have been new
champions for mp-qs on a supercomputer if we had kept that category.
Silverman found two new ecm champions: the p32 of 6,167+ and the p31 of
5,176+. This p32 is about half as large as the current p-1 champion
(the p32 of 2,977-). The p39 of 6,106+ whose digits are given on Page
45 is a new record penultimate factor.  The other p39 of 6,106+ is
175787157418305877173455355755546870641, which is just a bit larger.
Does anyone see an algebraic factorization here?

Atkin factored the number 1223165341640099735851, which is listed in
the book as a 22-digit prime factor of 6,175-. When he reported this
error, I performed a primality test on each of the more than 10,000
factors whose digits are given in the tables. They are all prime except
the one that Atkin factored. They include all primes below 2446. The
error happened between 1978 and 1981. The two p11 factors were probably
found together by the p-1 method. The largest prime factor of each
p11-1 is between 45,000 and 50,000.

Silverman finished the c70's on Page 45. Now the smallest entry in
Appendix C has 71 digits.

Gostin reports new factors of F_n  for n = 61, 64, 122, 142 and 906.
I will save them for Update # 4.


Keep the factors coming!


Sam Wagstaff