(124812 · ((599098 · ((135704 · ((25370 · ((50352 · ((58764 · ((2380 · ((1680 · ((5010 · ((1056 · ((1870 · ((170 · ((60 · (191804507173048342 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1

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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:(124812 · ((599098 · ((135704 · ((25370 · ((50352 · ((58764 · ((2380 · ((1680 · ((5010 · ((1056 · ((1870 · ((170 · ((60 · (191804507173048342 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1))2 + 1
Verification status (*):Proven
Official Comment (*):Generalized Fermat
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):p269 : Zhou, OpenPFGW
Decimal Digits:304031   (log10 is 304030.57398022)
Rank (*):22326 (digit rank is 1)
Entrance Rank (*):419
Currently on list? (*):no
Submitted:11/22/2010 14:03:49 UTC
Last modified:3/11/2023 15:54:10 UTC
Removed (*):9/2/2013 13:37:36 UTC
Database id:96548
Status Flags:none
Score (*):42.9763 (normalized score 0.1342)

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 Fermat (archivable *)
Prime on list: no, rank 4409
Subcategory: "Generalized Fermat"
(archival tag id 212992, tag last modified 2024-04-24 19:37:19)

User comments about this prime (disclaimer):

User comments are allowed to convey mathematical information about this number, how it was proven prime.... See our guidelines and restrictions.

Lei Zhou writes (11 Sep 2014):  (report abuse)
Details please see HERE

Proof code:
#!/bin/sh
./pfgw -l"GF.19.cert" -t -h"p_03" p_04
./pfgw -l"GF.19.cert" -t -h"p_04" p_05
./pfgw -l"GF.19.cert" -t -h"p_05" p_06
./pfgw -l"GF.19.cert" -t -h"p_06" p_07
./pfgw -l"GF.19.cert" -t -h"p_07" p_08
./pfgw -l"GF.19.cert" -t -h"p_08" p_09
./pfgw -l"GF.19.cert" -t -h"p_09" p_10
./pfgw -l"GF.19.cert" -t -h"p_10" p_11
./pfgw -l"GF.19.cert" -t -h"p_11" p_12
./pfgw -l"GF.19.cert" -t -h"p_12" p_13
./pfgw -l"GF.19.cert" -t -h"p_13" p_14
./pfgw -l"GF.19.cert" -t -h"p_14" p_15
./pfgw -l"GF.19.cert" -t -h"p_15" p_16
./pfgw -l"GF.19.cert" -t -h"p_16" p_17
./pfgw -l"GF.19.cert" -t -h"p_17" p_18

Helpers:
p_03: ((101)*10)^2+1
p_04: ((1020101)*96)^2+1
p_05: ((((1020101)*96)^2+1)*2)^2+1
p_06: ((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1
p_07: ((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1
p_08: ((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1
p_09: ((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1
p_10: ((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1
p_11: ((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1
p_12: ((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1
p_13: ((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1
p_14: ((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1
p_15: ((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1
p_16: ((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1
p_17: ((((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1

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.
fieldvalue
prime_id96548
person_id9
machineDitto P4 P4
whattrial_divided
notesPFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] 1+(124812*(1+(599098*(1+(.........2))^2))^2))^2))^ 1/1 trial factoring to 111499735 1+(124812*(1+(599098*(1+(135704*(1+(25370*(1+(50352*(1+(58764*(1+(2380*(1+(1680*(1+(5010*(1+(1056*(1+(1870*(1+(170*(1+(60*(1+(2*(97929696^2+1))^2))^2))^2))^2))^2))^2))^2))^2))^2))^2))^2))^2))^2))^2 has no small factor. [Elapsed time: 2420.192 seconds]
modified2020-07-07 22:30:33
created2010-11-22 14:05:02
id122482

fieldvalue
prime_id96548
person_id9
machineRedHat Virtual STEM Server
whatprime
notesPFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14] Primality testing (124812*((599098*((135704*((25370*((50352*((58764*((2380*((1680*((5010*((1056*((1870*((170*((60*((2*(97929696^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1 [N-1, Brillhart-Lehmer-Selfridge] Reading factors from helper file helper Prime_Testing_Warning, unused factor from helper file: 1020101 Prime_Testing_Warning, unused factor from helper file: 9590225358652417 Running N-1 test using base 5 Calling Brillhart-Lehmer-Selfridge with factored part 100.00% (124812*((599098*((135704*((25370*((50352*((58764*((2380*((1680*((5010*((1056*((1870*((170*((60*((2*(97929696^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1))^2+1 is prime! (9560.0506s+0.0512s) [Elapsed time: 2.66 hours] Helper File: 1020101 ((1020101)*96)^2+1 ((((1020101)*96)^2+1)*2)^2+1 ((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1 ((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1 ((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1 ((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1 ((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1 ((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1 ((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1 ((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1 ((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1 ((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1 ((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1 ((((((((((((((((((((((((((((1020101)*96)^2+1)*2)^2+1)*60)^2+1)*170)^2+1)*1870)^2+1)*1056)^2+1)*5010)^2+1)*1680)^2+1)*2380)^2+1)*58764)^2+1)*50352)^2+1)*25370)^2+1)*135704)^2+1)*599098)^2+1
modified2020-07-07 22:30:33
created2010-11-27 18:59:25
id122601

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