July 1, 1988


On Page 52, "AKL + MM" means "A.K. Lenstra and M. Manasse". They
factored three "More Wanted" numbers, namely, 10,116+ c94, 10,137-
c101 and 2,706M c88, all by ECM. Using the Quadratic Sieve on a NEC
SX-2, te Riele factored the "More Wanted" number 7,122+ c87 and the
"Most Wanted" number 6,131- c92. Brent found a 27-digit factor of
the "More Wanted" number 11,128+ c118 by ECM, leaving a c92, which is
still "More Wanted". There are new "Wanted" lists in Update 2.2, which
is enclosed. Silverman is factoring 3,194+ c89 now.

When I prepared the cover letter for Page 51 in April, I forgot to
mention that Colquitt and Welsch discovered a new Mersenne prime,
namely, M_110503 . Slowinski has checked this prime.


The most unexpected factorization of 1988 has to be the complete
factorization of the eleventh Fermat number by Brent. Two very small
factors of F_11 were known for decades. Brent found two more small
factors by ECM, and noted that the remaining cofactor was a probable
prime with 564 digits. Morain then proved that this cofactor is indeed
prime. Thus, F_11 has a total of five prime factors.


Using Atkin's method, Morain has finished the primality proof for
2,1093+ p315. He has also proved that all large probable primes found
recently actually are prime. Odlyzko has continued to supply primality
proofs for all Cunningham probable primes up to 210 digits, using A. K.
Lenstra's program. At this point, for the first time, prime proofs have
been provided for all the probable primes in the Cunningham Project
(through Update 2.2)!

The first holes which were done on Page 52 are those numbered 2497,
2507, 2508, 2517, 2532 and 2545. The only second hole factored on Page
52 was that in # 2494. Numbers 2510 and 2529 report the factorizations
of third holes. One fifth hole (in # 2525) was factored, too.

The smallest number in Appendix C presently has 83 digits. There are
just five c83's left and no 83-digit number was factored on Page 52.

There were new champions for both ECM and MP-QS on Page 52. Lenstra and
Manasse found the 35-digit ECM champion in # 2529 and the 36-digit ECM
champion in # 2545. The new MP-QS champion is the 92-digit number in
# 2508, factored by te Riele. In # 2517, Silverman found a new largest
penultimate prime factor, one with 42 digits. The latest list of
champions appears overleaf.

There is a new factorer on Page 52. He is Kenji Koyama of NTT. We wish
him success and look forward to receiving many factors from him.

Silverman has implemented an FFT second step for the Pollard p - 1
method. With it, he found several new factors on Page 52.

The smallest new factor on Page 52 has 15 digits. See # 2542.

In addition to the numbers which they factored on Page 52, Lenstra and
Manasse have made a substantial effort (thousands of curves) with ECM
on the following numbers: 2,419- c84, 2,445- c103, 2,449- c95,
2,368+ c93, 12,121- c94 and 2,361+ c103. Their Step 1 limit varied
from 320000 to about 1000000.

It is likely that the c75 left in # 2548 will be finished by the time
you read this.

At Update 2.2 there were 2182 primes in Appendix A and 1611 composites
in Appendix C.

The latest guess is that the Second Edition will appear in July.

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


Keep the factors coming!


Sam Wagstaff