November 22, 1986


On Page 47, Bob Silverman factored 5,128+ c87, which was on the "Most
Wanted" list for more than a year. This factorization of the first hole
in the 5+ table was a new champion for mp-qs.  See the other side for
the list of current champions for various factoring methods and for the
latest "Wanted" lists.

The factorizations numbered 2184, 2227, 2232 and 2240 were of the
second holes in their tables. Those numbered 2205, 2206 and 2239 were
of third holes in their tables. te Riele did # 2232 and Silverman did
the other six. Silverman factored several fourth and fifth holes, too.
The factorization of 3,187+ c77 was done by te Riele on a Cyber 205.
To the best of my knowledge, this is the largest number ever factored
by mp-qs on a single computer. Silverman has factored several larger
numbers by mp-qs on a distributed system of computers.

The remaining c72's, c73's and c74's in Appendix C all were factored
on Page 47. Silverman did most of this work, but I helped with a few
c73's. The elliptic curve method with many curves had been tried
earlier on these numbers. The mp-qs factorizations show that even a
large effort with ecm can miss some factors in the 20 to 25 digit
range.

I made a little chart of the number of decimal digits in the smallest
new factor found on each of these Pages. This size was 11 digits on
virtually all of the first 40 pages. Recently, however, this number has
begun to climb. On the last ten pages, that is, on Pages 38 to 47, the
size was 11, 11, 12, 14, 13, 12, 14, 11, 17 and 16 digits. Perhaps we
have finally found all of the 11 and 12 digit factors in the Tables.

When I mailed Page 46 in September, I forgot to mention Baillie's
discovery of a new factor of the Fermat number F_12:

2,4096+ c1202 1256132134125569. c1187 Baillie p-1

In the notation of the book, page lx, and of Update # 4, page 3,
his new factor has k = 76668221077, n = 14 and m = 12, where k =
7.7.53.29521841. Four smaller factors of F_12 were known earlier.

There is news about F_20, too. Duncan Buell and Jeff Young used
Pepin's theorem to prove that it is composite. We now know that the
Fermat numbers F_m are composite for 5 <= m <= 21.


Keep the factors coming!


Sam Wagstaff