From Fibonacci and Base12 Factors to Brent Factors

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

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 2

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 3

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 4

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 5

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 6

Sizes of non-largest Fibonacci factors missed in 1986 • • 23 44 35 31 17 10 34 ----194 p 21 -p 24 p 25 -p 29 p 30 -p 34 p 35 -p 39 p 40 -p 44 p 45 -p 49 higher Microsoft Research, USA 7

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 8

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 9

2003 Cunningham Base-12 Table • 66 composite 12 n ± 1 where n 300. • 13 composite Aurifeuillians for 12 n + 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 10

33 Base-12 done by SNFS • 489 L, 489 M, 507 M, 519 L, 543 L, 549 M, 555 L (Leherbauer), 561 M, 573 L, 579 M, 591 M, 597 M • 179−, 193−, 197− (NFSNET), 199−, 215−, 235−, 267−, 297− • 202+, 206+, 208+, 209+, 214+, 217+, 218+, 238+, 242+, 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 11

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 p 50 parameters. – Bruce Dodson had run many curves too. – ECM was unsuccessful. • 12, 242+ factored as p 50. p 159 (SNFS). Microsoft Research, USA 12

8 ECM Partial Factorizations • • 12, 232+ 12, 269− 12, 271+ 12, 274+ 12, 289− 12, 299+ p 45. c 181 p 42. c 205 p 50. c 234 p 42. c 250 p 47. c 213 p 46. c 248 p 44. c 174 p 53. c 202 (Daminelli) (Dodson) (Doligez) Microsoft Research, USA 13

9 ECM Full Factorizations • • • 12, 227− 12, 241+ 12, 251− 12, 259− 12, 262+ 12, 268+ 12, 272+ 12, 278+ 12, 281+ p 53. p 144 p 43. p 133 p 45. p 176 p 53. p 155 p 45. p 236 p 39. p 217 p 39. p 206 p 42. p 231 p 46. p 254 (more wanted) (Dodson) Microsoft Research, USA 14

2 GNFS Factorizations • 12, 280+ • 12, 236+ • 12, 229+ c 141 = p 61. p 81 c 151 = p 61. p 91 c 158 (more wanted, underway) • Smallest remaining cofactors – – 12, 287+ 12, 253+ 12, 299− 12, 266+ c 158 c 169 c 174 c 179 Microsoft Research, USA 15

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

Restarting Step 1 after Kill • Most ECM jobs used p 60 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 17

Longer ECM Runs • GMP-ECM 6. 1 -beta 2 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 B 1 ≈ 109, aiming for 65 -digit factors. • Alas, no new Cunningham factors (any base). Microsoft Research, USA 18

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 19

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 20

Early March, 2006 Brent Runs • Ran five curves on each of 1615 cofactors, with p 60 parameters. • Sixteen p 36 -p 45 factors, not all new. • March 21 table update down to 1471 cofactors. Microsoft Research, USA 21

April 2006 Brent Runs • Jens Franke busy running NFS jobs. • Five Pentiums jobs each, with p 60 parameters. • GMP-ECM 6. 1 -beta 2 released. – GMP-ECM 6. 0. 1 had often aborted on IA-64. – New code appeared stable. – About ten 6. 0. 1 p 60 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 22

May 2006 Brent Runs • Ten Pentium p 60 jobs per cofactor. – Found 26 ECM factors, 37 -47 digits. – One cofactor completed by GNFS. • Also one p 65 job for each c 200 and up. – Found p 41 and p 45. Microsoft Research, USA 23

June, 2006 • Started 12, 229+ c 158 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 24