Phi(631, - 21930489)

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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(631, - 21930489)
Verification status (*):	Proven
Official Comment (*):	Generalized unique
Proof-code(s): (*):	D : Dubner, Cruncher
Decimal Digits:	4625 (log₁₀ is 4624.8604388054)
Rank (*):	96083 (digit rank is 1)
Entrance Rank (*):	1230
Currently on list? (*):	no
Submitted:	7/1996
Last modified:	2/4/2026 11:56:27 UTC
Database id:	26240
Status Flags:	none
Score (*):	30.0468 (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 1557
Subcategory: "Generalized Unique"
(archival tag id 225601, 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 26240

person_id 9

machine Using: Digital Ocean Droplet

what prime

notes Command: /var/www/clientpool/1/pfgw64 -V -f -t -hhelper_1100000001857734606.txt -q"Phi(631,-21930489)" >command_output 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing Phi(631,-21930489) [N-1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file helper_1100000001857734606.txt
Prime_Testing_Warning,

Running N-1 test using base 23
Generic modular reduction using generic reduction FMA3 FFT length 1536 on A 15364-bit number
Calling Brillhart-Lehmer-Selfridge with factored part 33.48%

Phi(631,-21930489) is prime! (0.8252s+0.0052s)
[Elapsed time: 5.00 seconds]

Helper File:
2
3
5
7
11
13
19
29
31
37
43
61
71
109
127
139
151
181
199
211
223
379
421
449
487
631
757
991
1061
1171
1429
1621
2129
2341
2521
4481
4789
5881
6301
9241
17011
20071
33301
42841
89797
129061
131581
140551
145381
196201
200341
266221
338581
413713
462673
2320741
2652941
3083221
3325211
4097843
4460359
4670441
6345721
19339559
21925807
26962921
28842367
34099381
46898377
380291941
1980415627
4261628911
9755202151
26915598781
27939636217
1484386063021
3519025164181
12310216188859
15517711941751
233145031836221
390448344015073
390520145218681
480946325848633
872384493917623
2574437659606751
112752398888225671
467873735823752581
585348165239884957
1286624484128252791
107279048723166721801
4420366802492656686049
10534548204193953207211
1453145325510536919879391
1646084217547988735842951
12596690599264997413213242253
30260490515931052636700607109
574275812333239648724577270521
95144504120567541417400312916966041
571178649134691354741861153175459288381
825378916731728378352656906717483395621
141551955664713442005646495053219011773980899449801
159205032976511074156057993093371045510508573473058201
303594319896076968607621451899814051536787158631301595231
53504036083775628216533922805371415301934666848137276057441
378525666044715670293882984966402073270791642840872623379669607663786025849
16781379052073264101248662674423086...(107 digits)...51730120676177061266252705079492527
20218814180578287414181947282194448...(120 digits)...25771910013977927471180117426928941
28749139029971768165915701739611693...(129 digits)...06049159970838276953346214200829301
48891968304454639045149238181908596...(162 digits)...82514999526083023596239545696569351
15719830526101352988491138911915258...(219 digits)...19186259478377999006171015857184481
701
977
2371
3187
47924761

modified 2026-02-04 11:56:27

created 2026-02-04 11:56:22

id 187736

field	value
prime_id	26240
person_id	9
machine	Using: Digital Ocean Droplet
what	prime
notes	Command: /var/www/clientpool/1/pfgw64 -V -f -t -hhelper_1100000001857734606.txt -q"Phi(631,-21930489)" >command_output 2>&1 PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11] Primality testing Phi(631,-21930489) [N-1, Brillhart-Lehmer-Selfridge] Reading factors from helper file helper_1100000001857734606.txt Prime_Testing_Warning, Running N-1 test using base 23 Generic modular reduction using generic reduction FMA3 FFT length 1536 on A 15364-bit number Calling Brillhart-Lehmer-Selfridge with factored part 33.48% Phi(631,-21930489) is prime! (0.8252s+0.0052s) [Elapsed time: 5.00 seconds] Helper File: 2 3 5 7 11 13 19 29 31 37 43 61 71 109 127 139 151 181 199 211 223 379 421 449 487 631 757 991 1061 1171 1429 1621 2129 2341 2521 4481 4789 5881 6301 9241 17011 20071 33301 42841 89797 129061 131581 140551 145381 196201 200341 266221 338581 413713 462673 2320741 2652941 3083221 3325211 4097843 4460359 4670441 6345721 19339559 21925807 26962921 28842367 34099381 46898377 380291941 1980415627 4261628911 9755202151 26915598781 27939636217 1484386063021 3519025164181 12310216188859 15517711941751 233145031836221 390448344015073 390520145218681 480946325848633 872384493917623 2574437659606751 112752398888225671 467873735823752581 585348165239884957 1286624484128252791 107279048723166721801 4420366802492656686049 10534548204193953207211 1453145325510536919879391 1646084217547988735842951 12596690599264997413213242253 30260490515931052636700607109 574275812333239648724577270521 95144504120567541417400312916966041 571178649134691354741861153175459288381 825378916731728378352656906717483395621 141551955664713442005646495053219011773980899449801 159205032976511074156057993093371045510508573473058201 303594319896076968607621451899814051536787158631301595231 53504036083775628216533922805371415301934666848137276057441 378525666044715670293882984966402073270791642840872623379669607663786025849 16781379052073264101248662674423086...(107 digits)...51730120676177061266252705079492527 20218814180578287414181947282194448...(120 digits)...25771910013977927471180117426928941 28749139029971768165915701739611693...(129 digits)...06049159970838276953346214200829301 48891968304454639045149238181908596...(162 digits)...82514999526083023596239545696569351 15719830526101352988491138911915258...(219 digits)...19186259478377999006171015857184481 701 977 2371 3187 47924761
modified	2026-02-04 11:56:27
created	2026-02-04 11:56:22
id	187736

field value

prime_id 26240

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 23 Primality testing Phi(631,-21930489) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 47 Running N-1 test using base 53 Running N+1 test using discriminant 67, base 27+sqrt(67) Calling N-1 BLS with factored part 3.76% and helper 0.28% (11.57% proof) Phi(631,-21930489) is Fermat and Lucas PRP! (392.560000 seconds)

modified 2003-03-25 17:22:55

created 2003-01-05 07:04:25

id 62236

field	value
prime_id	26240
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 23 Primality testing Phi(631,-21930489) [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 47 Running N-1 test using base 53 Running N+1 test using discriminant 67, base 27+sqrt(67) Calling N-1 BLS with factored part 3.76% and helper 0.28% (11.57% proof) Phi(631,-21930489) is Fermat and Lucas PRP! (392.560000 seconds)
modified	2003-03-25 17:22:55
created	2003-01-05 07:04:25
id	62236

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