April 15, 1988


On Page 51, "AKL + MM" means "A.K. Lenstra and M. Manasse". There was
not enough room for both names. Near the top of Page 51, they factored
four "More Wanted" numbers, namely, 6,137+ c99, 7,137- c101, 12,106+
c99 and 5,157- c89. Silverman factored the "More Wanted" number
10,97- c89, which was the last number in the Cunningham Project
having exponent < 100.  Now only two numbers with base > 2 which were
listed in the Cunningham-Woodall tables of 1925 remain unfactored. They
are 10,109+ c93 and 11,107- c96. Both are "Most Wanted" now.

Update 2.1 appeared with much-needed new "Wanted" lists. Five of the
"More Wanted" numbers on the new list have already been done. Lenstra
and Manasse factored 2,361- c90, 6,143- c91 and 7,121+ c89.
Silverman factored 5,160+ c90 and Kida factored 3,223- c107. Update
2.2, which will appear next summer, will have new "Wanted" lists.

The first holes which were done on Page 51 are those numbered 2432,
2434, 2435, 2439, 2443, 2448, 2453, 2462, 2482, 2483 and 2487. The
second holes done on this page have numbers 2436, 2452, 2455, 2466,
2491 and 2492. Numbers 2463 and 2464 were the only third holes done.
Several fourth and fifth holes were factored, too.

On Page 51, Silverman factored the last c82's in the project. There are
just five c83's left.

There were new champions for both ECM and MP-QS on Page 51. Brent found
the 35-digit ECM champion in # 2482. Lenstra and Manasse found the
34-digit ECM champion in # 2439.  Silverman factored the two new MP-QS
champions, the 90-digit number in # 2483 and the 89-digit number in
# 2443. The former also has a new largest penultimate prime factor,
one with 41 digits.

In # 2467, Brent found the first known factor of 2^823 - 1.

Down to # 2470 we wondered whether the smallest new factor on Page 51
would have 22 or more digits. However, the p15 in # 2472 showed that
some small factors still lurk in the base 2 tables.

For several years, I have been hoping that someone would finish the
primality proofs for the large PRP's in Appendix A. In Update 2.1, I
reported that F. Morain proved primality of the prp222 of 2,1958M. I am
delighted to report that he has continued his work and has completed
the proofs for all of the PRP's except 2,1093+ prp315 and that he is
doing that last proof now.

Four new factors of Fermat numbers were reported in Update 2.1. Since
then, Gostin found one more, namely, the one with k = 497531, n = 859
and m = 851. We thank Convex Computer Corp. for supporting his work.

Buell and Young found a new largest non-Mersenne prime, namely,
8423.2^59877 + 1.

In addition to the numbers which they factored on Page 51, Lenstra and
Manasse have made a substantial effort (thousands of curves) with ECM
on the following numbers: All "Most wanted" numbers except 2,512+ and
5,256+; the "More wanted" numbers 3,194+, 11,113-, and 12,104+; and
the numbers 2,391-, 3,211+, and 6,142+. Their Step 1 limit varied
from 320000 to about 1000000.

I expect that the c68 left in # 2489 will be finished by the time you
read this.

The latest guess is that the Second Edition will appear in May or June.


Keep the factors coming!


Sam Wagstaff