mersenneforum.org  

Go Back   mersenneforum.org > Factoring Projects > Cunningham Tables

Closed Thread
 
Thread Tools
Old 2007-08-13, 17:14   #12
wblipp
 
wblipp's Avatar
 
"William"
May 2003
New Haven

45058 Posts
Default

Quote:
Originally Posted by bdodson View Post
This short table's now one shorter, after 7,317- c202 = p51*p151, with

p51 = 399676211948778347375576150888294069325575626336229
Just a nitpick. It's p51*p152, not p151.
wblipp is offline  
Old 2007-12-10, 09:11   #13
fivemack
(loop (#_fork))
 
fivemack's Avatar
 
Feb 2006
Cambridge, England

2·7·461 Posts
Default Factorisation of 7^263-1

87808581809525792134370860453721419342992604246001922594107781828847401497090250623000616403314204187748641525231019439257581826238153281895147142837427201101541 = 330356473356706630143084182110481076968694111519479821312366757981 * 265799489010506998952247476548157401788003618449001603472530395754819736985343973809633195038761

Software used: gnfs-lasieve4I14e for sieving, msieve-1.30 for linalg (the merge phase did not fit in 4GB under msieve-1.28)

Polynomials used: x^6-7 and 7^44-x

Sieving limits: small primes to 40 million, large primes to 2^30

Sieved algebraic special-Q from 36.5e6 to 79e6 in 5600 CPU-hours, on six CPUs (three dual-core machines) of average 2.3GHz

Found 90729983 relations and 45934 free relations, from which 82107167 unique relations

Constructed a 7163141 x 7163388 matrix of weight 478901308 in 173 minutes

Solved in 176 hours (two CPUs), using 2.5G memory

Square root took five hours; solution found on the first dependency

msieve log attached
Attached Files
File Type: txt 7-263.msieve.log.txt (15.7 KB, 289 views)
fivemack is offline  
Old 2007-12-10, 14:48   #14
R.D. Silverman
 
R.D. Silverman's Avatar
 
"Bob Silverman"
Nov 2003
North of Boston

748410 Posts
Default 7,263-

Nice.

What do you intend to do next?

I am about 85% sieved with 6,299- and will do 2,776+ next,
which will take a while with my resources. I get about 140 hrs/day
with my collection of machines.
R.D. Silverman is offline  
Old 2007-12-10, 15:21   #15
fivemack
(loop (#_fork))
 
fivemack's Avatar
 
Feb 2006
Cambridge, England

2×7×461 Posts
Default

At present I'm doing, with a collaborator, a C156 GNFS on a non-Cunningham number; I expect this to take about as much CPU time as 7,263-, but thanks to the collaboration there will be more CPUs working on it, and it might be finished by mid-January. I don't yet have a plan for the stage after that; I'm vaguely contemplating wider collaborations to do a larger GNFS job.
fivemack is offline  
Old 2007-12-12, 13:24   #16
akruppa
 
akruppa's Avatar
 
"Nancy"
Aug 2002
Alexandria

9A316 Posts
Default

I'll do 7,269- c224.

Alex
akruppa is offline  
Old 2008-01-10, 15:19   #17
bdodson
 
bdodson's Avatar
 
Jun 2005
lehigh.edu

210 Posts
Default

Quote:
Originally Posted by garo View Post
Code:
Base	Index	Size	11M(45digits)	43M(50digits)	110M(55digits)	260M(60digits)	Decimal
7	269-	C224	0(0.267423)	0(0.0522979)	165(0.00921839)	0(0.00148122)	1660265...
7	271-	C214	0(0.267423)	0(0.0522979)	165(0.00921839)	0(0.00148122)	157020...
7	277-	C201	0(0.290133)	30(0.0561584)	165(0.00980315)	0(0.00148122)	274499...
7	281-	C165	0(2.63512)	2160(0.477957)	0(0.0754361)	300(0.0068697)	
185248827501208343062750485931112817975629927609513170657610022785421928683373794979982241274691966304649468484472255815588739520597115203376303263535408573237014157
...
Looks like a 4th hole, pending Alex's breaking of the 1st hole. 7,281- c165
done with

p53 = 41601158252896546668861917466611604127922008421733517

found during the 5th t50, before 4.5*t50, but not exactly late for a p53.
2nd of two factors this morning, the other in the [still non-sticky] 2-
discussion. -Bruce
bdodson is offline  
Old 2008-01-14, 13:16   #18
akruppa
 
akruppa's Avatar
 
"Nancy"
Aug 2002
Alexandria

2,467 Posts
Default

7,269-:

Code:
Probable prime factor 1 has 82 digits:
4462060605403381961780411332028022754534242152003773412732171224503049883657394827
Probable prime factor 2 has 142 digits:
3720848769582826092237529016309237616548794511295400940058189854803271477562040904704041158521598520073343762354423669834498921937027750690971
Alex

Last fiddled with by akruppa on 2008-01-14 at 13:17
akruppa is offline  
Old 2008-01-14, 13:27   #19
fivemack
(loop (#_fork))
 
fivemack's Avatar
 
Feb 2006
Cambridge, England

2·7·461 Posts
Default

Congratulations! What are you sieving next?
fivemack is offline  
Old 2008-01-14, 13:31   #20
akruppa
 
akruppa's Avatar
 
"Nancy"
Aug 2002
Alexandria

1001101000112 Posts
Default

I'm not sieving anything at the moment, I'll run P+/-1 stage 2 on a bunch of residues we accumulated for a while. The matrix for 3,533- is at 80%. When I do NFS again, it'll probably be one or both of the c160 from the 3- table.

Alex
akruppa is offline  
Old 2008-02-25, 08:52   #21
fivemack
(loop (#_fork))
 
fivemack's Avatar
 
Feb 2006
Cambridge, England

11001001101102 Posts
Default

7^387-1 C154 cofactor splits as
Code:
P70 4583142681916056752995023804494031799489400087334962946866717203441543
P84 457299108883653825883853223306465080566360541916826721338777118806091728304473122859
Not quite sure how much CPU time I used, but it took two weeks real-time on maybe six cores; 100 hours for the linear algebra (95 hours of that on two cores for the matrix), using msieve-1.29 with default parameters because I'd just had a couple of failures with msieve-1.33. x^6+x^3+1, x-7^43; 30-bit large primes, small primes <25e6 on both sides.

Code:
Wed Feb 20 21:58:34 2008  Msieve v. 1.29
Wed Feb 20 21:58:34 2008  random seeds: 81975669 db274a81
Wed Feb 20 21:58:34 2008  factoring 2095867064326852049468708025219963891883437656350890684735312307910868519605941526543012489340565259400896935298036531088556378430157620037769819463531437 (154 digits)
Wed Feb 20 21:58:34 2008  commencing number field sieve (154-digit input)
Wed Feb 20 21:58:35 2008  R0:  2183814375991796599109312252753832343
Wed Feb 20 21:58:35 2008  R1: -1
Wed Feb 20 21:58:35 2008  A0:  1
Wed Feb 20 21:58:35 2008  A1:  0
Wed Feb 20 21:58:35 2008  A2:  0
Wed Feb 20 21:58:35 2008  A3:  1
Wed Feb 20 21:58:35 2008  A4:  0
Wed Feb 20 21:58:35 2008  A5:  0
Wed Feb 20 21:58:35 2008  A6:  1
Wed Feb 20 21:58:35 2008  size score = 1.957443e-10, Murphy alpha = 1.935662, combined = 1.125911e-10
Wed Feb 20 22:09:03 2008  restarting with 86633601 relations
Wed Feb 20 22:09:03 2008  
Wed Feb 20 22:09:03 2008  commencing relation filtering
Wed Feb 20 22:09:03 2008  commencing duplicate removal, pass 1
Wed Feb 20 22:09:17 2008  error -10 reading relation 1722289
Wed Feb 20 22:14:11 2008  error -10 reading relation 37387593
Wed Feb 20 22:15:08 2008  error -10 reading relation 44262255
Wed Feb 20 22:18:31 2008  error -10 reading relation 68351484
Wed Feb 20 22:21:32 2008  found 18462022 hash collisions in 86633597 relations
Wed Feb 20 22:21:32 2008  commencing duplicate removal, pass 2
Wed Feb 20 22:30:05 2008  found 7991052 duplicates and 78642545 unique relations
Wed Feb 20 22:30:05 2008  memory use: 504.8 MB
Wed Feb 20 22:30:28 2008  ignoring smallest 3249249 rational and 3248797 algebraic ideals
Wed Feb 20 22:30:28 2008  filtering ideals above 54412471
Wed Feb 20 22:30:28 2008  need 11046678 more relations than ideals
Wed Feb 20 22:30:28 2008  commencing singleton removal, pass 1
Wed Feb 20 22:42:48 2008  relations with 0 large ideals: 2150491
Wed Feb 20 22:42:48 2008  relations with 1 large ideals: 11700615
Wed Feb 20 22:42:48 2008  relations with 2 large ideals: 26144693
Wed Feb 20 22:42:48 2008  relations with 3 large ideals: 27165833
Wed Feb 20 22:42:48 2008  relations with 4 large ideals: 11291370
Wed Feb 20 22:42:48 2008  relations with 5 large ideals: 189543
Wed Feb 20 22:42:48 2008  relations with 6 large ideals: 0
Wed Feb 20 22:42:48 2008  relations with 7+ large ideals: 0
Wed Feb 20 22:42:48 2008  78642545 relations and about 57801182 large ideals
Wed Feb 20 22:42:48 2008  commencing singleton removal, pass 2
Wed Feb 20 22:54:41 2008  found 17689591 singletons
Wed Feb 20 22:54:41 2008  current dataset: 60952954 relations and about 38361402 large ideals
Wed Feb 20 22:54:47 2008  commencing singleton removal, pass 3
Wed Feb 20 23:04:39 2008  relations with 0 large ideals: 2150491
Wed Feb 20 23:04:39 2008  relations with 1 large ideals: 10507928
Wed Feb 20 23:04:39 2008  relations with 2 large ideals: 21103317
Wed Feb 20 23:04:39 2008  relations with 3 large ideals: 19705929
Wed Feb 20 23:04:39 2008  relations with 4 large ideals: 7370522
Wed Feb 20 23:04:39 2008  relations with 5 large ideals: 114767
Wed Feb 20 23:04:39 2008  relations with 6 large ideals: 0
Wed Feb 20 23:04:39 2008  relations with 7+ large ideals: 0
Wed Feb 20 23:04:39 2008  60952954 relations and about 48513054 large ideals
Wed Feb 20 23:04:39 2008  commencing singleton removal, pass 4
Wed Feb 20 23:14:26 2008  found 13709829 singletons
Wed Feb 20 23:14:26 2008  current dataset: 47243125 relations and about 33452791 large ideals
Wed Feb 20 23:14:30 2008  commencing singleton removal, pass 5
Wed Feb 20 23:22:40 2008  found 3591646 singletons
Wed Feb 20 23:22:40 2008  current dataset: 43651479 relations and about 29746456 large ideals
Wed Feb 20 23:22:45 2008  commencing singleton removal, pass 6
Wed Feb 20 23:30:58 2008  found 946498 singletons
Wed Feb 20 23:30:58 2008  current dataset: 42704981 relations and about 28791073 large ideals
Wed Feb 20 23:31:02 2008  commencing singleton removal, pass 7
Wed Feb 20 23:39:15 2008  found 247491 singletons
Wed Feb 20 23:39:15 2008  current dataset: 42457490 relations and about 28542982 large ideals
Wed Feb 20 23:39:19 2008  commencing singleton removal, final pass
Wed Feb 20 23:48:14 2008  memory use: 1315.2 MB
Wed Feb 20 23:48:14 2008  commencing in-memory singleton removal
Wed Feb 20 23:48:19 2008  begin with 42457490 relations and 32642470 unique ideals
Wed Feb 20 23:49:32 2008  reduce to 35506412 relations and 25489736 ideals in 17 passes
Wed Feb 20 23:49:32 2008  max relations containing the same ideal: 44
Wed Feb 20 23:49:36 2008  dataset has 54.1% excess relations
Wed Feb 20 23:49:50 2008  ignoring smallest 2942925 rational and 2942497 algebraic ideals
Wed Feb 20 23:49:50 2008  filtering ideals above 48971223
Wed Feb 20 23:49:50 2008  need 9015008 more relations than ideals
Wed Feb 20 23:49:50 2008  commencing singleton removal, final pass
Wed Feb 20 23:59:34 2008  memory use: 1307.2 MB
Wed Feb 20 23:59:34 2008  commencing in-memory singleton removal
Wed Feb 20 23:59:39 2008  begin with 42457490 relations and 33254381 unique ideals
Thu Feb 21 00:00:55 2008  reduce to 35486599 relations and 26081020 ideals in 17 passes
Thu Feb 21 00:00:55 2008  max relations containing the same ideal: 47
Thu Feb 21 00:01:20 2008  removing 2175672 relations and 1980387 ideals in 195285 cliques
Thu Feb 21 00:01:22 2008  commencing in-memory singleton removal
Thu Feb 21 00:01:26 2008  begin with 33310927 relations and 26081020 unique ideals
Thu Feb 21 00:02:02 2008  reduce to 33228356 relations and 24017486 ideals in 9 passes
Thu Feb 21 00:02:02 2008  max relations containing the same ideal: 45
Thu Feb 21 00:02:25 2008  removing 1572631 relations and 1377346 ideals in 195285 cliques
Thu Feb 21 00:02:26 2008  commencing in-memory singleton removal
Thu Feb 21 00:02:30 2008  begin with 31655725 relations and 24017486 unique ideals
Thu Feb 21 00:03:00 2008  reduce to 31609120 relations and 22593236 ideals in 8 passes
Thu Feb 21 00:03:00 2008  max relations containing the same ideal: 44
Thu Feb 21 00:03:22 2008  removing 12223 relations and 11348 ideals in 875 cliques
Thu Feb 21 00:03:22 2008  commencing in-memory singleton removal
Thu Feb 21 00:03:26 2008  begin with 31596897 relations and 22593236 unique ideals
Thu Feb 21 00:03:33 2008  reduce to 31596896 relations and 22581887 ideals in 2 passes
Thu Feb 21 00:03:33 2008  max relations containing the same ideal: 44
Thu Feb 21 00:03:43 2008  dataset has 38.7% excess relations
Thu Feb 21 00:03:55 2008  ignoring smallest 2634707 rational and 2634397 algebraic ideals
Thu Feb 21 00:03:55 2008  filtering ideals above 43529976
Thu Feb 21 00:03:55 2008  need 7212007 more relations than ideals
Thu Feb 21 00:03:55 2008  commencing singleton removal, final pass
Thu Feb 21 00:11:51 2008  memory use: 1069.9 MB
Thu Feb 21 00:11:51 2008  commencing in-memory singleton removal
Thu Feb 21 00:11:54 2008  begin with 31596896 relations and 23196884 unique ideals
Thu Feb 21 00:12:20 2008  reduce to 31581170 relations and 23181158 ideals in 7 passes
Thu Feb 21 00:12:20 2008  max relations containing the same ideal: 45
Thu Feb 21 00:12:43 2008  removing 2552970 relations and 2152970 ideals in 400000 cliques
Thu Feb 21 00:12:45 2008  commencing in-memory singleton removal
Thu Feb 21 00:12:48 2008  begin with 29028200 relations and 23181158 unique ideals
Thu Feb 21 00:13:15 2008  reduce to 28892502 relations and 20890951 ideals in 8 passes
Thu Feb 21 00:13:15 2008  max relations containing the same ideal: 43
Thu Feb 21 00:13:35 2008  removing 2301750 relations and 1901750 ideals in 400000 cliques
Thu Feb 21 00:13:37 2008  commencing in-memory singleton removal
Thu Feb 21 00:13:40 2008  begin with 26590752 relations and 20890951 unique ideals
Thu Feb 21 00:14:01 2008  reduce to 26472647 relations and 18869598 ideals in 7 passes
Thu Feb 21 00:14:01 2008  max relations containing the same ideal: 42
Thu Feb 21 00:14:19 2008  removing 2080170 relations and 1689129 ideals in 391041 cliques
Thu Feb 21 00:14:21 2008  commencing in-memory singleton removal
Thu Feb 21 00:14:24 2008  begin with 24392477 relations and 18869598 unique ideals
Thu Feb 21 00:14:43 2008  reduce to 24287430 relations and 17074041 ideals in 7 passes
Thu Feb 21 00:14:43 2008  max relations containing the same ideal: 40
Thu Feb 21 00:14:59 2008  removing 15591 relations and 14210 ideals in 1381 cliques
Thu Feb 21 00:15:00 2008  commencing in-memory singleton removal
Thu Feb 21 00:15:03 2008  begin with 24271839 relations and 17074041 unique ideals
Thu Feb 21 00:15:08 2008  reduce to 24271837 relations and 17059829 ideals in 2 passes
Thu Feb 21 00:15:08 2008  max relations containing the same ideal: 40
Thu Feb 21 00:15:15 2008  dataset has 11.0% excess relations
Thu Feb 21 00:15:26 2008  ignoring smallest 2324114 rational and 2324293 algebraic ideals
Thu Feb 21 00:15:26 2008  filtering ideals above 38088729
Thu Feb 21 00:15:26 2008  need 7017889 more relations than ideals
Thu Feb 21 00:15:26 2008  commencing singleton removal, final pass
Thu Feb 21 00:22:52 2008  memory use: 835.6 MB
Thu Feb 21 00:22:52 2008  commencing in-memory singleton removal
Thu Feb 21 00:22:54 2008  begin with 24271837 relations and 17678440 unique ideals
Thu Feb 21 00:23:14 2008  reduce to 24251083 relations and 17657685 ideals in 7 passes
Thu Feb 21 00:23:14 2008  max relations containing the same ideal: 40
Thu Feb 21 00:23:16 2008  dataset has 1.5% excess relations
Thu Feb 21 00:23:19 2008  relations with 0 large ideals: 696101
Thu Feb 21 00:23:19 2008  relations with 1 large ideals: 3279608
Thu Feb 21 00:23:19 2008  relations with 2 large ideals: 7156515
Thu Feb 21 00:23:19 2008  relations with 3 large ideals: 7840821
Thu Feb 21 00:23:19 2008  relations with 4 large ideals: 4313963
Thu Feb 21 00:23:19 2008  relations with 5 large ideals: 964075
Thu Feb 21 00:23:19 2008  relations with 6 large ideals: 0
Thu Feb 21 00:23:19 2008  relations with 7+ large ideals: 0
Thu Feb 21 00:23:19 2008  commencing 2-way merge
Thu Feb 21 00:23:44 2008  reduce to 15318012 relation sets and 8724614 unique ideals
Thu Feb 21 00:23:44 2008  commencing full merge
Thu Feb 21 00:28:56 2008  found 7372243 cycles, need 5562814
Thu Feb 21 00:29:03 2008  weight of 5562814 cycles is about 361703750 (65.02/cycle)
Thu Feb 21 00:29:03 2008  distribution of cycle lengths:
Thu Feb 21 00:29:03 2008  1 relations: 1064006
Thu Feb 21 00:29:03 2008  2 relations: 712935
Thu Feb 21 00:29:03 2008  3 relations: 615306
Thu Feb 21 00:29:03 2008  4 relations: 540088
Thu Feb 21 00:29:03 2008  5 relations: 489752
Thu Feb 21 00:29:03 2008  6 relations: 444477
Thu Feb 21 00:29:03 2008  7 relations: 405935
Thu Feb 21 00:29:03 2008  8 relations: 368677
Thu Feb 21 00:29:03 2008  9 relations: 330499
Thu Feb 21 00:29:03 2008  10+ relations: 591139
Thu Feb 21 00:29:03 2008  heaviest cycle: 13 relations
Thu Feb 21 00:29:04 2008  commencing cycle optimization
Thu Feb 21 00:29:15 2008  start with 26668809 relations
Thu Feb 21 00:30:17 2008  pruned 674553 relations
Thu Feb 21 00:30:19 2008  distribution of cycle lengths:
Thu Feb 21 00:30:19 2008  1 relations: 1064006
Thu Feb 21 00:30:19 2008  2 relations: 730569
Thu Feb 21 00:30:19 2008  3 relations: 638531
Thu Feb 21 00:30:19 2008  4 relations: 554043
Thu Feb 21 00:30:19 2008  5 relations: 507268
Thu Feb 21 00:30:19 2008  6 relations: 458475
Thu Feb 21 00:30:19 2008  7 relations: 418951
Thu Feb 21 00:30:19 2008  8 relations: 373663
Thu Feb 21 00:30:19 2008  9 relations: 325551
Thu Feb 21 00:30:19 2008  10+ relations: 491757
Thu Feb 21 00:30:19 2008  heaviest cycle: 13 relations
Thu Feb 21 00:30:26 2008  
Thu Feb 21 00:30:26 2008  commencing linear algebra
Thu Feb 21 00:30:35 2008  read 5562814 cycles
Thu Feb 21 00:31:09 2008  cycles contain 15512158 unique relations
Thu Feb 21 00:38:17 2008  read 15512158 relations
Thu Feb 21 00:38:49 2008  using 32 quadratic characters above 1073741312
Thu Feb 21 00:47:45 2008  read 5562814 cycles
Thu Feb 21 00:50:44 2008  filtering completed in 3 passes
Thu Feb 21 00:50:46 2008  matrix is 5409318 x 5409509 with weight 467914660 (avg 86.50/col)
Thu Feb 21 00:52:34 2008  read 5409509 cycles
Thu Feb 21 00:52:48 2008  matrix is 5409318 x 5409509 with weight 467914660 (avg 86.50/col)
Thu Feb 21 00:52:48 2008  saving the first 48 matrix rows for later
Thu Feb 21 00:52:51 2008  matrix is 5409270 x 5409509 with weight 356348373 (avg 65.87/col)
Thu Feb 21 00:52:51 2008  matrix includes 64 packed rows
Thu Feb 21 00:52:51 2008  using block size 65536 for processor cache size 4096 kB
Thu Feb 21 00:53:25 2008  commencing Lanczos iteration (2 threads)
Mon Feb 25 00:02:25 2008  lanczos halted after 85543 iterations (dim = 5409234)
Mon Feb 25 00:02:39 2008  recovered 35 nontrivial dependencies
Mon Feb 25 00:02:41 2008  
Mon Feb 25 00:02:41 2008  commencing square root phase
Mon Feb 25 00:02:41 2008  reading relations for dependency 1
Mon Feb 25 00:02:49 2008  read 2702718 cycles
Mon Feb 25 00:03:01 2008  cycles contain 8998845 unique relations
Mon Feb 25 00:11:58 2008  read 8998845 relations
Mon Feb 25 00:13:24 2008  multiplying 12692472 relations
Mon Feb 25 00:51:21 2008  multiply complete, coefficients have about 308.42 million bits
Mon Feb 25 00:51:29 2008  initial square root is modulo 342047
Mon Feb 25 01:35:00 2008  prp70 factor: 4583142681916056752995023804494031799489400087334962946866717203441543
Mon Feb 25 01:35:00 2008  prp84 factor: 457299108883653825883853223306465080566360541916826721338777118806091728304473122859
Mon Feb 25 01:35:00 2008  elapsed time 99:36:26
fivemack is offline  
Old 2008-06-23, 19:11   #22
Raman
Noodles
 
Raman's Avatar
 
"Mr. Tuch"
Dec 2007
Chennai, India

3×419 Posts
Thumbs up Not an ECM miss, certainly worth by SNFS

7,295-

Code:
Tue Jun 24 00:30:50 2008  prp81 factor: 204239004182680605398190478754212368873366912490836010105265524712426411236134031
Tue Jun 24 00:30:51 2008  prp111 factor: 393263672474017252292660491631044385409360056708958704520879019006886885032467377758314801669636946200575798561
One minute...
Let me mail Prof. Sam Wagstaff before posting further information about it...
Raman is offline  
Closed Thread

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
7+ table garo Cunningham Tables 87 2022-03-25 19:16
5+ table garo Cunningham Tables 100 2021-01-04 22:36
6+ table garo Cunningham Tables 80 2021-01-04 22:33
5- table garo Cunningham Tables 82 2020-03-15 21:47
6- table garo Cunningham Tables 41 2016-08-04 04:24

All times are UTC. The time now is 12:04.


Fri Oct 7 12:04:03 UTC 2022 up 50 days, 9:32, 0 users, load averages: 1.00, 1.15, 1.15

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2022, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.

≠ ± ∓ ÷ × · − √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °
∠ ∟ ° ≅ ~ ‖ ⟂ ⫛
≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘ ∏ ∐ ∑ ∧ ∨ ∩ ∪ ⨀ ⊕ ⊗ 𝖕 𝖖 𝖗 ⊲ ⊳
∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟
¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣ … ⋯ ⋮ ⋰ ⋱
∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ
𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎𝜍 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔