September 17, 1985 Only three people said that they did not want to receive the pages of New factors, like the enclosed Page 31. Three is so small compared to the size of the mailing list that I decided to continue to send the New factors to everyone who receives the annual Updates. It is clear from Page 31 that the multiple polynomial quadratic sieve (mp-qs) and the elliptic curve method (ecm) have enjoyed great success lately. Eighteen of the 28 factorizations on the "wanted" lists of Update # 3 have been done. Silverman did six of the Ten Most Wanted and 7 of the Eighteen More Wanted. Montgomery did four of the More Wanted. Atkin and Rickert did two Most and two More Wanted numbers. Some of these numbers were factored twice by different methods. The first "holes" of many tables were filled in. All of the More Wanted Factorizations published on page lix of the book have been obtained. (The Ten Most Wanted ones were all done more than a year ago.) Here are the new wanted lists which John Selfridge and I just prepared: Ten Most Wanted Factorizations 1. 2,512+ c148 6. 7,128+ c95 2. 2,454L c68 7. 2,272+ c74 3. 10,73+ c70 8. 6,97+ c75 4. 5,128+ c87 9. 10,97- c89 5. 6,128+ c75 10. 10,89+ c86 Seventeen More Wanted Factorizations 2,277- c78 3,193- c72 7,89- c69 2,289- c75 3,149+ c71 7,89+ c65 2,259+ c66 5,109+ c72 10,83- c70 2,269+ c81 6,113- c66 11,79- c79 2,502M c72 6,94+ c69 11,79+ c69 2,518L c66 12,79+ c85 It was reported in today's Los Angeles Times that David Slowinski has found a new Mersenne prime. The exponent is 216091. Bob Baillie reports that Slowinski has tested all exponents up to that point, so that M(216091) really is the thirtieth Mersenne prime. The new largest prime is easy to remember since 216 is 6 cubed and 91 is the product of the first two primes congruent to 1 modulo 6. While writing this letter, I received these two entries for page 32: 1140 5, 131- c 73 16815642611861. p60 Dubner BP-rho 1141 2, 299- c 60 13444476836590589479.51441563151591093599. p21 Wunderlich cfrac(2) Keep the factors coming! Sam Wagstaff