From Fibonacci and Base-12 Factors to Brent Factors

1. From Fibonacci and Base-12 Factors to Brent Factors Peter L. Montgomery Microsoft Research and CWI Microsoft Research, USA

2. Abstract • After I wrote factoring codes in 1983, I focused on Fibonacci Fn and Lucas Ln cofactors. Three of us published those tables in 1988, with many gaps. Both tables were complete for n 1000 by 2003. I switched to base-12, and cut its composite cofactor list in half. Now that the “easy” base-12 have been done, I am transitioning to Brent’s tables. Microsoft Research, USA

3. Ein Rätsel • Exclusive OR the letters in BRENT + U + ANTS • Remove repeats • Insert other duplicates • Rearrange • As in BRENT + ERECT = CANBERRA What fitting 10-letter word can we get? Microsoft Research, USA

4. CWI “Factory” Jobs • Run nights and weekends in Amsterdam • Typical hours 18:30 to 08:00. • My awake hours in western USA. • Only machines where owner consents. • Owner can kill factory job at any time. • ECM and NFS jobs • 80-100 machines usable 80-100 hours/week. • Mostly AMD Athlons • Forty more machines added June, 2006. • Also have budget at nearby SARA, on IA-64’s. Microsoft Research, USA

5. Fibonacci and Lucas tables • 1988 Math. Comp. paper with John Brillhart and Robert Silverman had Fibonacci factorizations Fn (n 1000) and Lucas factorizations Ln (n 500). • 140 Fn and 10 Ln incomplete (out of 1050 entries). • By April, 2003, both complete for n 1000. • http://blair.kelly.home.att.net/mathematics/fibonacci/ Microsoft Research, USA

6. How 150 factorizations completed • 53 ECM • 51 SNFS • 43 MPQS • 2 P−1/FFT (Silverman) • 1 GNFS (Kleinjung) ----- 150 Many had earlier ECM or P−1/FFT factor. Microsoft Research, USA

7. Sizes of non-largest Fibonacci factors missed in 1986 • 23 p21-p24 • 44 p25-p29 • 35 p30-p34 • 31 p35-p39 • 17 p40-p44 • 10 p45-p49 • 34 higher ----- 194 Microsoft Research, USA

8. Fibonacci Contributors 1988-2003 • Richard Brent • Nicolas Daminelli • Marije Elkenbracht-Huizing • Jens Franke • Sean Irvine • Thorsten Kleinjung • Joe Leherbauer • Mark Manasse (MPQS network) • Robert Silverman • Thomas Sosnowski I used systems at Unisys, UCLA, Oregon State, CWI, SARA. Microsoft Research, USA

9. 2003 -- Time to Switch Tables • Cunningham table has factors of bn± 1 for b = 2, 3, 5, 6, 7, 10, 11, 12. • Had worked on this intermittently since 1984. • Brent table has higher bases. • Old August,1999 copy had 1224 composites. • Factors of bn ± 1 where 13 b 29 and bn< 10255, or 30 b 99 and n 100. • Too many “easy” SNFS (difficulty circa 10110). Experienced labor overload in 1999. Microsoft Research, USA

10. 2003 Cunningham Base-12 Table • 66 composite 12n ± 1 where n 300. • 13 composite Aurifeuillians for 12n + 1 where n≡ 3 (mod 6) and n < 600. • Total 79 composites March, 2003. • 35 composites as of June, 2006. • First hole advanced from 12,179− to 12,227+. Microsoft Research, USA

11. 33 Base-12 done by SNFS • 489L, 489M, 507M, 519L, 543L, 549M, 555L (Leherbauer), 561M, 573L, 579L, 579M, 591M, 597M • 179−, 193−, 197− (NFSNET), 199−, 215−, 235−, 267−, 297− • 202+, 206+, 208+, 209+, 214+, 217+, 218+, 238+, 242+, 282+, 288+, 300+ • Larger exponents divisible by 3, 5, 7, or 11. • Easiest SNFS left have difficulty 12212 for 265+ (degree 4) and 12222 for 259+ (degree 6). Microsoft Research, USA

12. Don’t Neglect ECM • 12,242+ sieving estimated to take over six calendar months (m = 1222 + 12−22). • Meanwhile (mid-2005) Bruce Dodson finds record-size factors using new GMP-ECM 6.0.1. • Interrupted SNFS runs to run 5000 ECM curves with p50 parameters. • Bruce Dodson had run many curves too. • ECM was unsuccessful. • 12,242+ factored as p50.p159 (SNFS). Microsoft Research, USA

13. 8 ECM Partial Factorizations • 12,232+ p45.c181 (Daminelli) • 12,269− p42.c205 • 12,271− p50.c234 • 12,271+ p42.c250 • 12,274+ p47.c213 (Dodson) • 12,289− p46.c248 • 12,299− p44.c174 (Doligez) • 12,299+ p53.c202 Microsoft Research, USA

14. 9 ECM Full Factorizations • 12,227− p53.p144 (more wanted) • 12,241+ p43.p133 (Dodson) • 12,251− p45.p176 • 12,259− p53.p155 (Dodson) • 12,262+ p45.p236 • 12,268+ p39.p217 • 12,272+ p39.p206 • 12,278+ p42.p231 • 12,281+ p46.p254 Microsoft Research, USA

15. 2 GNFS Factorizations • 12,280+ c141 = p61.p81 • 12,236+ c151 = p61.p91 • 12,229+ c158 (more wanted, underway) • Smallest remaining cofactors • 12,287+ c158 • 12,253+ c169 • 12,299− c174 • 12,266+ c179 Microsoft Research, USA

16. Three Years of Base-12 Work • 33 completed by SNFS • 9 completed by ECM • 2 completed by GNFS • 1 GNFS running now • 8 ECM factor found, composite cofactor • 26 No progress ----- 79 Microsoft Research, USA

17. Restarting Step 1 after Kill • Most ECM jobs used p60 parameters. • Perhaps four hours step 1, one hour step 2. • Factory jobs are killed at 08:00 each weekday. • CWI NFS code checkpoints every 30 minutes as well as when SIGTERM received. • GMP-ECM 6.0.1 can save/resume between step 1 and step 2, but not within step 1. • If running step 1 at 08:00, waste an average two hours when it restarts at 18:30. • Waste grows as step 1 duration increases. Microsoft Research, USA

18. Longer ECM Runs • GMP-ECM 6.1-beta2 supports checkpoint within step 1 when SIGTERM received. • Now we worry much less about having jobs complete before 08:00. • Trying some ECM jobs with B1 ≈ 109, aiming for 65-digit factors. • Alas, no new Cunningham factors (any base). Microsoft Research, USA

19. Time for Another Table • Why switch? • Little 2006 progress in base-12 table. • Annual CWI report showed few 2005-2006 results (any table), after long 12,242+ effort. • By returning to Brent’s table, we will have more results to report next year. • Started transition in February, 2006. Microsoft Research, USA

20. Brent Table from 1999 to 2005 • Latest Brent table update, from September, 2005, had 1615 composites. • Up from 1224 composites in 1999. • Table had been extended to have all bn±1 where 13 b 99 and bn< 10255 . • Silverman et al had finished all n 100. • Most easy SNFS cofactors from 1999 had been done, but some easy exponents divisible by 21 remained. Microsoft Research, USA

21. Early March, 2006 Brent Runs • Ran five curves on each of 1615 cofactors, with p60 parameters. • Sixteen p36-p45 factors, not all new. • March 21 table update down to 1471 cofactors. Microsoft Research, USA

22. April 2006 Brent Runs • Jens Franke busy running NFS jobs. • Five Pentiums jobs each, with p60 parameters. • GMP-ECM 6.1-beta2 released. • GMP-ECM 6.0.1 had often aborted on IA-64. • New code appeared stable. • About ten 6.0.1 p60 curves per cofactor at SARA. • Overall got 33 ECM factors, 35-51 digits. • Four cofactors completed by GNFS. • April 22 table update had 1420 composites. Microsoft Research, USA

23. May 2006 Brent Runs • Ten Pentium p60 jobs per cofactor. • Found 26 ECM factors, 37-47 digits. • One cofactor completed by GNFS. • Also one p65 job for each c200 and up. • Found p41 and p45. Microsoft Research, USA

24. June, 2006 • Started 12,229+ c158 by GNFS June 17. • 40 new hosts added to factory pool – two calendar months estimated sieving time. • 73 M relations as of July 17 – 80% of goal. Microsoft Research, USA

25. Was ist die Antwort? • BRENT + U + ANTS = ?? • A binary anagram • 10-letter word Microsoft Research, USA

26. Antwort • BRENT + U + ANTS = GEBURTSTAG • HAPPY BIRTHDAY, RICHARD! Microsoft Research, USA