May 18, 1989 On Page 56, "AKL + MM" means "A.K. Lenstra and M. Manasse". They factored the "More Wanted" numbers 2,379+ c101 and 5,166+ c108 by ECM. With the assistance of dozens of other researchers, they factored the "Most Wanted" number 2,353+ c106 and set a new record for factoring integers by MP-QS. The first holes done on Page 56 are listed in # 2766 and # 2805. No second holes were factored on Page 56. The only third hole done on Page 56 is in # 2796. No fourth holes were factored on Page 56. The only fifth hole done on Page 56 is in # 2781. A list of the current first four holes in the base 2 and base 5 tables is enclosed. (The first four holes of the other tables have not changed since the previous report.) The Cunningham authors use the term "tiny" to mean an important unfactored number which is too small to be "Wanted". We are happy when someone factors one of these unglamorous numbers. We are grateful to Bob Silverman for factoring the "tinies" numbered 2781 and 2800. The latter, 7,259L c87, was the last remaining higher-base Aurifueillian from the first edition. Silverman found a 38-digit prime factor of the Fibonacci number U467 c98 by ECM. This is a new record for ECM, although it is not in the Cunningham Project. The base 11 tables have been extended again. The extension is given on an enclosed sheet. I am extending the base 12 tables now. The factorizations of 3,591M c94, 3,537L c86 and 3,549M c86 on Page 56 finish the base 3 L's and M's published in the second edition. It seems that the base 3 tables will need to be extended next. Silverman finished the last five c86's. There are two c87's and one c88 left in Appendix C. I hope they get done on Page 57. (There are no c89's left at present. Silverman factored the last one, 5,365L.) While testing a NeXT computer, Richard Crandall factored 2,4093- c1233. Although this number is not in the book, its factorization will interest many readers. Aside from the factor of 2,4093-, the smallest new factor reported on Page 56 has 17 digits. See # 2765. It is the very last base 2 number listed in the book and its factorization is now complete. The c336 reported in # 2772 is the largest number from the Cunningham Project to be factored since # 1654, that is, for more than three years. Eight Cunningham numbers with more than 300 digits were factored on Page 56. For the first time since the base 2 tables were extended to 1200, the number of Cunningham composites with more than 300 digits has dropped below 100. There are 97 of them left now. Update 2.3 will have new wanted lists. It will appear in late June or early July. I have corrected several of your addresses recently. If you move, please tell me. Keep the factors coming! Sam Wagstaff