214114 - 3
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: | 214114 - 3 |
|---|---|
| Verification status (*): | Proven |
| Official Comment (*): | [none] |
| Proof-code(s): (*): | F : Forbes |
| Decimal Digits: | 4249 (log10 is 4248.7373588014) |
| Rank (*): | 96848 (digit rank is 2) |
| Entrance Rank (*): | 1448 |
| Currently on list? (*): | no |
| Submitted: | 5/1996 |
| Last modified: | 2/4/2026 12:18:39 UTC |
| Removed (*): | 3/4/1998 13:31:14 UTC |
| Database id: | 27027 |
| Status Flags: | none |
| Score (*): | 29.7831 (normalized score 0) |
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 27027 person_id 9 machine Using: Digital Ocean Droplet what prime notes Command: /var/www/clientpool/1/pfgw64 -V -f -tc -hhelper_1100000000294459053.txt -q"2^14114-3" >command_output 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing 2^14114-3 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file helper_1100000000294459053.txt
trial
Running N-1 test using base 2
Special modular reduction using FMA3 FFT length 768 on 2^14114-3
Running N-1 test using base 11
Special modular reduction using FMA3 FFT length 768 on 2^14114-3
Running N-1 test using base 23
Special modular reduction using FMA3 FFT length 768 on 2^14114-3
Running N+1 test using discriminant 31, base 1+sqrt(31)
Special modular reduction using FMA3 FFT length 768 on 2^14114-3
Calling N-1 BLS with factored part 52.21% and helper 1.01% (157.67% proof)
2^14114-3 is prime! (0.4443s+0.0001s)
[Elapsed time: 5.00 seconds]
Helper File:
2
3
5
7
13
17
19
29
37
43
73
97
109
113
127
193
197
241
257
337
433
449
577
673
883
1009
1153
1429
2017
2689
3137
3361
3529
5153
5419
6337
7057
14449
21169
22051
34273
38737
50177
65537
84673
92737
101921
126127
258721
273617
309583
311347
373969
540961
649657
689921
748819
1007441
1608769
1693441
1994497
2627857
5828257
15790321
22253377
47886721
85225897
183076097
269389009
375327457
423319681
720636337
2015814529
19707683773
25629623713
38941695937
40388473189
42303768193
54410972897
77158673929
88959882481
118750098349
278452876033
487824887233
1405628248417
4363953127297
4432676798593
4981857697937
7150715595073
358429848460993
364565561997841
1538595959564161
3040410389842561
66308056470365249
26032885845392093851
40544859693521152369
1475204679190128571777
5091542821206332688289
469721803307970049499713
2365315148221200764324737
2741672362528725535068727
4487533753346305838985313
16059450203006241829176289
17059410504738323992180849
494077391563970390335488001
1996187656530838599012839257
23811637619463293269503628033
169462032913464877812492288268723
7086423574853972147970086088434689
7201935329692363728844274296784241409
7026820612597192347604169969096107647937
624220395843565891905204297675049041529729
265549217634074770386573489863592827112366481
1006545610655423007817780018765739801419395841
131084304485119425504284495119889529996019181850241
6225705473971244034925408772395577890626984089129137
21566694047156190030012960957098417375263164101421409
163129078976626145477798423051187262197756524727184748929
10032718675660700331163223888781708908159501730496624539621810378577
84998535361121926317825290336868414402467054590903922313327679343257
14510642956629460126286667764218111732339625499480335264478327629658324054225616417
33725933170854542422930854135636663761123331227599520852573351371057495391835351809
58693755727744825443995323040350236...(113 digits)...28292652490231165838210929789110401
42834275261569859429440376549803331...(180 digits)...45537933945813554489408221027542209
10438573647812762612333343129211438...(259 digits)...36921413695633172682060575234858097
23
89
4824675346114250541198242904214396192319modified 2026-02-04 12:18:39 created 2026-02-04 12:18:34 id 187739
field value prime_id 27027 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 2 Running N-1 test using base 11 Running N+1 test using discriminant 23, base 3+sqrt(23) Primality testing 2^14114-3 [N-1/N+1, Brillhart-Lehmer-Selfridge] Calling N-1 BLS with factored part 4.68% and helper 0.08% (14.14% proof) 2^14114-3 is Fermat and Lucas PRP! (151.300000 seconds) modified 2003-03-25 17:22:56 created 2003-01-05 05:02:38 id 62145
Query times: 0.0002 seconds to select prime, 0.0003 seconds to seek comments.
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