June 23, 1993 This mailing contains Page 68, Update 2.7 and an extension of the tables. Three "Most Wanted" numbers were factored on Page 68. From the wanted lists mailed with Page 66, Bruce Dodson factored 2,509+ c150 and 10,157- c132 using the Elliptic Curve Method. Paul Leyland factored 12,131+ c136 using ECM. Nine "More Wanted" numbers were factored on Page 68. Bob Silverman factored 2,814L c109, 2,838L c116, 2,842L c110, 3,283+ c114, 3,293+ c122 and 5,187- c106, all by the Number Field Sieve. Bruce Dodson factored 6,179- c115 by the Elliptic Curve Method. Zayer et al. factored 3,278+ c107 by the Number Field Sieve. The group "FactOregon" factored 3,295+ c111 by the Number Field Sieve. New wanted lists may be found in Update 2.7. Bob Silverman factored one of the "Smaller but Needed" numbers, 5,187+ c105, by the Number Field 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. Nine of these numbers were factored on Page 68. Silverman factored 3,283+, 3,293+, 5,187- and 5,187+ by NFS. He also factored 6,191+ by ECM. Zayer et al. factored 3,278+ by NFS. Dodson factored 6,179- by ECM. The group "FactOregon" factored 3,295+ by NFS. Leyland factored 12,131+ by ECM. At this writing, 24 of these numbers remain to be factored. All of them appear on the "Wanted" lists except the two (5,193+ and 3,329+) whose factorization is in progress. Only seven numbers with b^n smaller than 10^140 remain unfinished. All are higher base first edition numbers. 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 penultimate prime factor is the p61 of 3,293+. A list of recent champions and the first holes in each table is given on another sheet. NFS champions are the ones with largest values of b^n +- 1 because that is the size of the number which must be factored. When the new General NFS begins to factor Cunningham numbers we will create a new category of champions for it. The size of its champions will be the size of the number actually factored, as for QS. The abbreviation AKL+MM means Arjen K. Lenstra and Mark Manasse. FactOregon is an abbreviation for Joe Buhler, Scott Huddleston, Peter Montgomery, Robby Robson and Russell Ruby. We use mpecm to refer to an Elliptic Curve program for the MasPar computer written by Arjen Lenstra and Brandon Dixon. The first holes done on Page 68 are in # 3504, 3519, 3521 3524, 3526, 3528, 3537, 3539, 3540, 3541 and 3546. The second holes done on Page 68 are in # 3487, 3511 and 3517. The only third hole done on Page 68 is in # 3508. No fourth holes were done on Page 68. The only fifth hole done on Page 68 is in # 3499. The smallest new factor reported on Page 68 has 24 digits. See # 3533. Only 14 numbers smaller than 100 digits remain in Appendix C. There are six c97's and four each with 98 and 99 digits. A c97 was added when the base 5 table was extended from 270 to 280. I have corrected several of your addresses recently. If you move, please tell me. Keep the factors coming! Sam Wagstaff