March 1, 1993

Three "Wanted" numbers were factored on Page 67. From the new wanted
lists mailed with Page 66, Bruce Dodson found factors of the "More
Wanted" numbers 7,169+ c128 and 2,515+ c123 using the Elliptic Curve
Method. The latter left a c89 cofactor which Arjen Lenstra finished by
the Quadratic Sieve. I factored the "More Wanted" number 5,191+ c114
using the ECM program of Lenstra and Dixon on Purdue's MasPar computer.
New wanted lists will be issued with Update 2.7 next summer.

Bob Silverman factored five of the "Smaller but Needed" numbers,
6,169+ c101, 12,127+ c99, 6,172+ c99, 2,830L c99 and 2,854L c98,
all by the Quadratic Sieve. The latest list of "Smaller but Needed"
numbers appears on the "Champions" page.

Our current goal is to factor all the higher base (b > 2) numbers
listed in the first (1983) edition of our book. Seven of these numbers
were factored on Page 67. Bruce Dodson found factors of 7,169+, 5,208+,
6,188+, 6,191+ and 6,194+, all by ECM. He left a c89 cofactor of
5,208+, which Arjen Lenstra finished by the Quadratic Sieve, and a c109
cofactor of 6,191+, which remains unfinished. Arjen Lenstra, Mark
Manasse and the network factored 5,202+ by the Quadratic Sieve. I
factored 5,191+ c114 using the ECM program of Lenstra and Dixon. At
this writing, 33 of these numbers remain to be factored.

There is one new champion for factoring Cunningham numbers on this
page. Recall that a champion is one of the best two records in its
class. The new champion ECM factorization is the p39 of 3,415- c145. A
list of recent champions and the first holes in each table is given on
another sheet. Because we missed an NFS champion in the report with
Page 66, we list the best five NFS records here. NFS champions are the
ones with largest values of b^n +- 1.

The abbreviation AKL means Arjen K. Lenstra. FactOregon is an
abbreviation for Joe Buhler, Scott Huttleston, Peter Montgomery, Robby
Robson and Russell Ruby. mpecm refers to an Elliptic Curve program for
the MasPar computer written by Arjen Lenstra and Brandon Dixon. ecmfft
is an Elliptic Curve program with a Fast Fourier Transform second step
written by Peter Montgomery.

The first holes done on Page 67 are in # 3429, 3433 and 3437. The
second holes done on Page 67 are in # 3439, 3442, 3451, 3462 and 3463.
The third holes done on Page 67 are in # 3444, 3453 and 3458. The
fourth holes done on Page 67 are in # 3457, 3465 and 3475.  The fifth
holes done on Page 67 are in # 3421, 3435, 3438, 3441, 3445, 3446 and
3455.

The smallest new factor reported on Page 67 has 19 digits. See # 3434.
This number is from the extension of the tables. The smallest new
factor of a number not from the extension has 24 digits and appears in
# 3472.

Only 13 numbers smaller than 100 digits remain in Appendix C. There are
five c97's and four each with 98 and 99 digits.

Harvey Dubner reports another new Fermat factor. He found that
19.2^23290 + 1 is prime and divides F_23288 . It is the new second
largest known factor of a Fermat number.

I have corrected several of your addresses recently. If you move,
please tell me.

Keep the factors coming!


Sam Wagstaff