August 3, 1991

Before I bring you the latest developments that relate to the
Cunningham Project, let me mention here that D. H. Lehmer passed away
in Berkeley on May 22, 1991, at age 86.  He was a coauthor and good
friend. He will be sorely missed by his many friends and colleagues.

Another avid reader of this newsletter, Sam Yates, also died recently.
He collected and published very large primes and factors of repunits.
Professor Chris Caldwell has agreed to continue this work. Please send
new primes > 10^1000  and repunit factors to him at Department of
Mathematics and Computer Science, University of Tennessee at Martin,
Martin, TN 38238.

Many "Wanted" numbers were factored on Page 63. From the wanted lists
of Update 2.4, Bob Silverman factored the "Most Wanted" numbers 2,428+ c114,
12,118+ c110 and 3,257- c118 and the "More Wanted" numbers 3,227- c98,
2,722M c103 and 2,445- c103, all by the Number Field Sieve. Arjen
Lenstra and Mark Manasse, with the help of dozens of other people and
computers, factored the "Most Wanted" number 10, 151- c113 and the
"More Wanted" numbers 2,499+ c104, 2,461+ c102, 2,460+ c106,
2,443+ c105, 2,439+ c105 and 2,479+ c108, all by the Quadratic Sieve
algorithm. Update 2.5 will have new wanted lists.

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. Factorizations of numbers in this category on Page 63
are shown in # 3202, 3205, 3207, 3213, 3218, 3226, 3238 and 3240. Only
eight of these numbers remain unfinished. All eight appear on the
wanted lists of Update 2.4 (shown on a separate sheet).

There were two new champions for factoring Cunningham numbers on this
page.  (Recall that a champion is one of the best TWO records in its
class.) The records were the factorization of 10,151- c113 by the
Quadratic Sieve and the penultimate factor of the same factorization. A
list of recent champions and first holes in each table is given on
another sheet.

The first holes done on Page 63 are in # 3187, 3196, 3202, 3213, 3220,
3225, 3226, 3237, 3238, 3239 and 3240. The only second hole done on
Page 63 is in # 3218. The third holes done on Page 63 are in # 3204 and
3207. No fourth holes were done on Page 63. The fifth holes done on
Page 63 are in # 3180 and 3191.

The smallest new factor reported on Page 63 has 21 digits. See # 3179,
3197 and 3233.

Bob Silverman factored the last 92-digit number from Appendix C on
March 15, 1991.  Arjen Lenstra and I factored the last six 93-digit
numbers. The last one was done on June 5, 1991. (On July 29, 1991,
Richard Brent left a new c93 in # 3235.) Arjen Lenstra factored the
last 94-digit number on May 17, 1991. He and Yuji Kida have been
working on the last 95-digit numbers. At present only one of them
remains.

Richard Crandall of NeXT Computer, Inc., reports finding another factor
of the Fermat number F_13 . It is 3603109844542291969 and has the same
number of digits (19) as the factor 2663848877152141313 of the same
Fermat number which he found and which was reported with Page 62. The
new cofactor is composite.

At the request of several readers I have decided to issue a memo
explaining the Pages and Updates once a year. It will be sent to you
with Update 2.5 in a few weeks.

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

Keep the factors coming!


Sam Wagstaff