Phi(631, - 13719616)

Previous Next Random

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(631, - 13719616)
Verification status (*):	Proven
Official Comment (*):	Generalized unique
Proof-code(s): (*):	D : Dubner, Cruncher
Decimal Digits:	4497 (log₁₀ is 4496.5254322925)
Rank (*):	96466 (digit rank is 1)
Entrance Rank (*):	1472
Currently on list? (*):	no
Submitted:	7/1996
Last modified:	2/4/2026 12:03:55 UTC
Database id:	26694
Status Flags:	none
Score (*):	29.9593 (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 1559
Subcategory: "Generalized Unique"
(archival tag id 225603, 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 26694

person_id 9

machine Using: Digital Ocean Droplet

what prime

notes Command: /var/www/clientpool/1/pfgw64 -V -f -t -hhelper_1100000001977775700.txt -q"Phi(631,-13719616)" >command_output 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing Phi(631,-13719616) [N-1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file helper_1100000001977775700.txt
Prime_Testing_Warning,

Running N-1 test using base 47
Generic modular reduction using generic reduction FMA3 FFT length 1536 on A 14938-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 33.35%

Phi(631,-13719616) is prime! (0.8685s+0.0051s)
[Elapsed time: 5.00 seconds]

Helper File:
2
3
5
7
11
13
19
23
29
31
37
43
61
71
73
109
127
181
211
271
307
379
421
463
577
631
757
1171
2129
2521
2591
2857
12097
14431
28621
47251
56101
58771
94781
319147
325081
694261
899851
1266301
4571971
6603211
7697971
8057221
23583691
98975341
187451881
561839041
1043440051
6112673491
9122727781
21605346241
22497917941
24877502791
96137135137
110367776801
122315212111
315851662981
561124086181
831661737761
1079336962243
4548193922791
8684640354931
188227849467841
188278694368441
197258347058221
812399741255881
3087610137018191
3893353230403883
11006020816631821
71236564035921901
102795913100000821
6811918194206452363
2582414003483249794561
631534658877771134268781
48072329133874344646126831
671822234466710977989338581
4343382359234524479179366341
12957899950167099799576017661
35420163219343947707020492681
1043932915759330872961784740181
2420694000042602484701883145969
67659551636472521309469865998131
234061923832743342436140146026873
108013501547459360097228425904141006181
930347319917997466467189783477488019612481
6670662533719740801764297729033556459724921
5940751095900225684743202455064729873970154311
553103859720264743303686313223175023346832580331777342109977021
202817488001378466154304706488172918869988487042532981187972003211681
18669249981917314039837805459788297...(99 digits)...14590937876621193523528484885467771
27478825014439952328917638745346178...(117 digits)...03104441612138207327514895919067627
69792679681855612659570328031712727...(149 digits)...48073854563307802572136048131782241
37193336849796227804634702535767516...(323 digits)...80355359583236455767706531634496101
7492097

modified 2026-02-04 12:03:55

created 2026-02-04 12:03:50

id 187737

field	value
prime_id	26694
person_id	9
machine	Using: Digital Ocean Droplet
what	prime
notes	Command: /var/www/clientpool/1/pfgw64 -V -f -t -hhelper_1100000001977775700.txt -q"Phi(631,-13719616)" >command_output 2>&1 PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11] Primality testing Phi(631,-13719616) [N-1, Brillhart-Lehmer-Selfridge] Reading factors from helper file helper_1100000001977775700.txt Prime_Testing_Warning, Running N-1 test using base 47 Generic modular reduction using generic reduction FMA3 FFT length 1536 on A 14938-bit number Calling Brillhart-Lehmer-Selfridge with factored part 33.35% Phi(631,-13719616) is prime! (0.8685s+0.0051s) [Elapsed time: 5.00 seconds] Helper File: 2 3 5 7 11 13 19 23 29 31 37 43 61 71 73 109 127 181 211 271 307 379 421 463 577 631 757 1171 2129 2521 2591 2857 12097 14431 28621 47251 56101 58771 94781 319147 325081 694261 899851 1266301 4571971 6603211 7697971 8057221 23583691 98975341 187451881 561839041 1043440051 6112673491 9122727781 21605346241 22497917941 24877502791 96137135137 110367776801 122315212111 315851662981 561124086181 831661737761 1079336962243 4548193922791 8684640354931 188227849467841 188278694368441 197258347058221 812399741255881 3087610137018191 3893353230403883 11006020816631821 71236564035921901 102795913100000821 6811918194206452363 2582414003483249794561 631534658877771134268781 48072329133874344646126831 671822234466710977989338581 4343382359234524479179366341 12957899950167099799576017661 35420163219343947707020492681 1043932915759330872961784740181 2420694000042602484701883145969 67659551636472521309469865998131 234061923832743342436140146026873 108013501547459360097228425904141006181 930347319917997466467189783477488019612481 6670662533719740801764297729033556459724921 5940751095900225684743202455064729873970154311 553103859720264743303686313223175023346832580331777342109977021 202817488001378466154304706488172918869988487042532981187972003211681 18669249981917314039837805459788297...(99 digits)...14590937876621193523528484885467771 27478825014439952328917638745346178...(117 digits)...03104441612138207327514895919067627 69792679681855612659570328031712727...(149 digits)...48073854563307802572136048131782241 37193336849796227804634702535767516...(323 digits)...80355359583236455767706531634496101 7492097
modified	2026-02-04 12:03:55
created	2026-02-04 12:03:50
id	187737

field value

prime_id 26694

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 47 Primality testing Phi(631,-13719616) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N+1 test using discriminant 59, base 15+sqrt(59) Calling N-1 BLS with factored part 2.85% and helper 0.01% (8.55% proof) Phi(631,-13719616) is Fermat and Lucas PRP! (262.900000 seconds)

modified 2003-03-25 17:22:55

created 2003-01-05 06:13:58

id 62201

field	value
prime_id	26694
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 47 Primality testing Phi(631,-13719616) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N+1 test using discriminant 59, base 15+sqrt(59) Calling N-1 BLS with factored part 2.85% and helper 0.01% (8.55% proof) Phi(631,-13719616) is Fermat and Lucas PRP! (262.900000 seconds)
modified	2003-03-25 17:22:55
created	2003-01-05 06:13:58
id	62201

Query times: 0.0003 seconds to select prime, 0.0005 seconds to seek comments.