December 16, 1991

Several "Wanted" numbers were factored on Page 64. From the old wanted
lists of Update 2.4 (November, 1990), Arjen Lenstra and Mark Manasse,
with the help of dozens of other people and computers, factored the
"More Wanted" number 2,481- c110 by the Quadratic Sieve algorithm.
Using the Number Field Sieve algorithm, Bob Silverman factored the
"More Wanted" numbers 5,167+ c104, 6,151- c105, 5,169- c105 and
3,263+ c105.

New wanted lists were issued with Update 2.5 last September and
Silverman factored the "More Wanted" numbers 6,164+ c125 and
6,163- c124 by NFS. He received some help in factoring these numbers
from David Willmore and from me. Lenstra, Manasse and the network are
now working on the c114 cofactor of 3,311-, which is "More Wanted".
Silverman is doing the "More Wanted" number 3,269+ c123 now.

Yuji Kida factored 7,163- c101, one of the "Smaller but Needed" numbers
mentioned in the cover letter for Update 2.5. I hope that other people
will try to factor the rest of these numbers, too.

The only numbers from the original 1925 Cunningham-Woodall tables which
have not yet been completely factored are a few base 2 numbers with
exponent < 500. One of these numbers, 2,481- c110, was factored on Page
64. Only seven of these numbers remain unfinished. All seven appear on
the wanted lists of Update 2.5.

There were no new champions for factoring Cunningham numbers on this
page. The first five holes in each table are listed on another sheet.

The first holes done on Page 64 are in # 3260, 3263, 3269, 3278, 3291,
3295 and 3300. The only second hole done on Page 64 is in # 3283. The
only third hole done on Page 64 is in # 3259. The only fourth hole done
on Page 64 is in # 3274. No fifth hole was done on Page 64.

The smallest new factor reported on Page 64 has 19 digits. See # 3282.
The number which was factored, 10,410M c138, comes from the recent
extension of the tables. These newly added numbers are more likely to
have a small prime factor. The smallest new factor reported on Page 64,
but not from a table extension, has 20 digits. See # 3275. This
factorization is interesting because it was done by the Quadratic Sieve
after hundreds of elliptic curves failed to find the 20-digit prime
factor.

Lenstra finished 3,331+ c93, which was left on Page 63. On August 10,
1991, he finished the last c95 from Appendix C. He and Kida finished
the last seven 96-digit numbers.  The last one was done on November 27,
1991. Eight 97-digit numbers are currently the smallest ones in
Appendix C.

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

Keep the factors coming!


Sam Wagstaff