May 14, 1986 Several exciting factorizations appear on Page 44. Let me begin at the top and work down the page. 2,301- c70 was the second hole in its table when Silverman factored it. He and Atkin independently factored 2,269+ c81, which was the first hole in its table. It was a 'most wanted' number and a new champion for QS. The current champions are listed at the end of this letter. Note that I have combined the two categories for QS. After 2,269+ was split, 2,293+ became the first hole and Montgomery factored the new second hole, 2,307+ c71. The current second hole in that table is 2,331+ c77. te Riele knocked off the last c64, 5,149+. Niebuhr got the 'more wanted' number 2,502M c72. Then Silverman did 2,277- c78, the first hole in its table. It was a new QS champion and on the 'most wanted' list. It also has the greatest known penultimate prime factor, a p38. Silverman finished the c69's and did a c68 which was left by Montgomery and Baillie (independently). At this moment the least number in Appendix C has 70 digits. An initial segment of the full Appendix C is shown on the three pages beginning on the back of Page 44. Yes, 5,139- is the last c84. When Silverman did 6,282M c72 he finished the base 6 Aurifeuillians. After 2,277- and 2,301- were done, 2,311- c87 and 2,313- c72 became the first two holes in their table. Then Brent factored the new third hole, 2,317- c74. Here is the latest list of champions. Note that #1777 has been added for ecm. I missed it earlier. Continued fraction method: 1692 C63 5,171+ McCurdy+Wunderlich on MPP 1602 C62 3,204+ Smith+Wagstaff on EPOC Quadratic sieve method: 1987 C81 2,269+ Silverman on a network of several SUN-3's 2037 C78 2,277- Silverman on a network of several SUN-3's SPAR: 1066 C60 3,131+ Atkin+Rickert on a large IBM 941 C59 12,59- Atkin+Rickert on a large IBM Pollard p-1: 921 P32 2,977- Brent on Univac 1100/82 719 P30 2,740+ Baillie on Cyber 130/730 p+1: 0 P21 2,439- Brent on Univac 1100/82 5 P20 2,509- Brent on Univac 1100/82 953 P19 2,456+ Suyama on a microcomputer. Brent-Pollard rho: 1343 P19 2,2386L Dubner on a special processor 1254 P19 2,1049- Dubner on a special processor Elliptic curve method: 1392 P29 5,199+ Silverman on VAX 1777 P28 10,83- Atkin+Rickert on a large IBM Keep the factors coming! Sam Wagstaff P.S. If you are moving soon, please send me your new address.