Phi(211, - 503941617438736)

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At this site we maintain a list of the 5000 Largest Known Primes which is updated hourly. This list is the most important PrimePages database: a collection of research, records and results all about prime numbers. This page summarizes our information about one of these primes.

This prime's information:

Description:	Phi(211, - 503941617438736)
Verification status (*):	Proven
Official Comment (*):	Generalized unique
Proof-code(s): (*):	D : Dubner, Cruncher
Decimal Digits:	3088 (log₁₀ is 3087.4998473649)
Rank (*):	99266 (digit rank is 1)
Entrance Rank (*):	2961
Currently on list? (*):	no
Submitted:	7/1996
Last modified:	2/4/2026 10:26:22 UTC
Database id:	29354
Status Flags:	none
Score (*):	28.79 (normalized score 0)

Archival tags:

There are certain forms classed as archivable: these prime may (at times) remain on this list even if they do not make the Top 5000 proper. Such primes are tracked with archival tags.

Generalized Unique (archivable *)
Prime on list: no, rank 1564
Subcategory: "Generalized Unique"
(archival tag id 225608, tag last modified 2026-01-09 22:37:30)

Verification data:

The Top 5000 Primes is a list for proven primes only. In order to maintain the integrity of this list, we seek to verify the primality of all submissions. We are currently unable to check all proofs (ECPP, KP, ...), but we will at least trial divide and PRP check every entry before it is included in the list.

field value

prime_id 29354

person_id 9

machine Using: Digital Ocean Droplet

what prime

notes Command: /var/www/clientpool/1/pfgw64 -V -f -t -hhelper_1100000002482408149.txt -q"Phi(211,-503941617438736)" >command_output 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing Phi(211,-503941617438736) [N-1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file helper_1100000002482408149.txt
Prime_Testing_Warning,

Running N-1 test using base 37
Generic modular reduction using generic reduction FMA3 FFT length 1K on A 10257-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 33.86%

Phi(211,-503941617438736) is prime! (0.2828s+0.0048s)
[Elapsed time: 5.00 seconds]

Helper File:
2
3
5
7
11
13
19
23
29
31
41
43
61
71
73
103
113
157
211
241
281
313
421
631
673
677
751
967
1579
2521
3541
5011
5279
10529
11467
17851
21841
28597
30493
88817
91841
252911
393919
670321
723997
3654751
6217681
10904221
936136681
1075580161
1681125241
1992985681
6268225867
9758474041
25500006757
536094353593
3388721700883
15618923846041
18783585101821
38764738076161
203574427577473
318847296013531
4037705916282361
4169778165215041
40464584955286451
45539207649630661
560717228692724491
4296391276974302101
26263175898324567217
164114610077326034281
270834555127246330921
11114585049094876966304689
24034388204576901152117056361
50791428494792777003986398761
144712931479794162850781148098941
1229927331488162750633762344843141177
4410933720963008000788300698078241201
127952567532321289443302601682149288239850679
6930799051255951569453226829526584371381976821
6163762754194934352328816112711043092507165285020976657693127473401
274012382773277213978629955245025199771070258749642797328999433967285945462835041
16378773329531821116568238102679265500832551248835063431223871355304597685132074274371781
15358108458658156935104405840386253...(98 digits)...53445487021702350619661722718110121
43154485336462033176845023153802721...(170 digits)...02753886013057271299053467452421751
32949808454278197242128017164718540...(349 digits)...28079211801825177314207431802265281
957091
2637163
86417797

modified 2026-02-04 10:26:22

created 2026-02-04 10:26:17

id 187690

field	value
prime_id	29354
person_id	9
machine	Using: Digital Ocean Droplet
what	prime
notes	Command: /var/www/clientpool/1/pfgw64 -V -f -t -hhelper_1100000002482408149.txt -q"Phi(211,-503941617438736)" >command_output 2>&1 PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11] Primality testing Phi(211,-503941617438736) [N-1, Brillhart-Lehmer-Selfridge] Reading factors from helper file helper_1100000002482408149.txt Prime_Testing_Warning, Running N-1 test using base 37 Generic modular reduction using generic reduction FMA3 FFT length 1K on A 10257-bit number Calling Brillhart-Lehmer-Selfridge with factored part 33.86% Phi(211,-503941617438736) is prime! (0.2828s+0.0048s) [Elapsed time: 5.00 seconds] Helper File: 2 3 5 7 11 13 19 23 29 31 41 43 61 71 73 103 113 157 211 241 281 313 421 631 673 677 751 967 1579 2521 3541 5011 5279 10529 11467 17851 21841 28597 30493 88817 91841 252911 393919 670321 723997 3654751 6217681 10904221 936136681 1075580161 1681125241 1992985681 6268225867 9758474041 25500006757 536094353593 3388721700883 15618923846041 18783585101821 38764738076161 203574427577473 318847296013531 4037705916282361 4169778165215041 40464584955286451 45539207649630661 560717228692724491 4296391276974302101 26263175898324567217 164114610077326034281 270834555127246330921 11114585049094876966304689 24034388204576901152117056361 50791428494792777003986398761 144712931479794162850781148098941 1229927331488162750633762344843141177 4410933720963008000788300698078241201 127952567532321289443302601682149288239850679 6930799051255951569453226829526584371381976821 6163762754194934352328816112711043092507165285020976657693127473401 274012382773277213978629955245025199771070258749642797328999433967285945462835041 16378773329531821116568238102679265500832551248835063431223871355304597685132074274371781 15358108458658156935104405840386253...(98 digits)...53445487021702350619661722718110121 43154485336462033176845023153802721...(170 digits)...02753886013057271299053467452421751 32949808454278197242128017164718540...(349 digits)...28079211801825177314207431802265281 957091 2637163 86417797
modified	2026-02-04 10:26:22
created	2026-02-04 10:26:17
id	187690

field value

prime_id 29354

person_id 9

machine Linux PII 200

what prp

notes PFGW Version 20020311.x86_Dev (Alpha software, 'caveat utilitor') Running N-1 test using base 37 Primality testing Phi(211,-503941617438736) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 53 Running N+1 test using discriminant 67, base 15+sqrt(67) Calling N-1 BLS with factored part 4.61% and helper 0.01% (13.86% proof) Phi(211,-503941617438736) is Fermat and Lucas PRP! (148.270000 seconds)

modified 2003-03-25 17:22:57

created 2003-01-04 22:19:37

id 61907

field	value
prime_id	29354
person_id	9
machine	Linux PII 200
what	prp
notes	PFGW Version 20020311.x86_Dev (Alpha software, 'caveat utilitor') Running N-1 test using base 37 Primality testing Phi(211,-503941617438736) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 53 Running N+1 test using discriminant 67, base 15+sqrt(67) Calling N-1 BLS with factored part 4.61% and helper 0.01% (13.86% proof) Phi(211,-503941617438736) is Fermat and Lucas PRP! (148.270000 seconds)
modified	2003-03-25 17:22:57
created	2003-01-04 22:19:37
id	61907

Query times: 0.0002 seconds to select prime, 0.0003 seconds to seek comments.