282904 · (547952 · (135236 · (20564 · (1752 · (306 · (112 · (1002 · (162 · (540 · (272 · (182 · (28 · 1572584048036110865273248812 + 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:282904 · (547952 · (135236 · (20564 · (1752 · (306 · (112 · (1002 · (162 · (540 · (272 · (182 · (28 · 1572584048036110865273248812 + 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 (*):[none]
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):p269 : Zhou, OpenPFGW
Decimal Digits:230336   (log10 is 230335.12530263)
Rank (*):28181 (digit rank is 1)
Entrance Rank (*):1089
Currently on list? (*):no
Submitted:8/15/2010 23:25:27 UTC
Last modified:3/11/2023 15:54:10 UTC
Removed (*):5/8/2012 17:09:06 UTC
Database id:94235
Status Flags:none
Score (*):42.1227 (normalized score 0.0567)

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"pmduo.20.cert" -t p_05
./pfgw -l"pmduo.20.cert" -tp -h"p_05" p_06
./pfgw -l"pmduo.20.cert" -t -h"p_06" p_07
./pfgw -l"pmduo.20.cert" -tp -h"p_07" p_08
./pfgw -l"pmduo.20.cert" -tp -h"p_08" p_09
./pfgw -l"pmduo.20.cert" -t -h"p_09" p_10
./pfgw -l"pmduo.20.cert" -t -h"p_10" p_11
./pfgw -l"pmduo.20.cert" -t -h"p_11" p_12
./pfgw -l"pmduo.20.cert" -t -h"p_12" p_13
./pfgw -l"pmduo.20.cert" -tp -h"p_13" p_14
./pfgw -l"pmduo.20.cert" -t -h"p_14" p_15
./pfgw -l"pmduo.20.cert" -tp -h"p_15" p_16
./pfgw -l"pmduo.20.cert" -tp -h"p_16" p_17
./pfgw -l"pmduo.20.cert" -tp -h"p_17" p_18
./pfgw -l"pmduo.20.cert" -t -h"p_18" p_19

Helpers:
p_05: (665857)^2*20-1
p_06: ((665857)^2*20-1)^2*2-1
p_07: (((665857)^2*20-1)^2*2-1)^2*28+1
p_08: ((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1
p_09: (((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1
p_10: ((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1
p_11: (((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1
p_12: ((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1
p_13: (((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1
p_14: ((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1
p_15: (((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1
p_16: ((((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1)^2*20564-1
p_17: (((((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1)^2*20564-1)^2*135236-1
p_18: ((((((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1)^2*20564-1)^2*135236-1)^2*547952-1

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_id94235
person_id9
machineDitto P4 P4
whattrial_divided
notesPFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] 282904*(547952*(135236*(2.........)^2+1)^2-1)^2-1) 1/1 trial factoring to 82985941 282904*(547952*(135236*(20564*(1752*(306*(112*(1002*(162*(540*(272*(182*(28*(2*8867310888979^2-1)^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 has no small factor. [Elapsed time: 1392.431 seconds]
modified2020-07-07 22:30:33
created2010-08-15 23:35:01
id117839

fieldvalue
prime_id94235
person_id9
machineRedHat Virtual STEM Server
whatprime
notesPFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14] Primality testing 282904*(547952*(135236*(20564*(1752*(306*(112*(1002*(162*(540*(272*(182*(28*(2*8867310888979^2-1)^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.94235 Running N-1 test using base 3 Calling Brillhart-Lehmer-Selfridge with factored part 100.00% 282904*(547952*(135236*(20564*(1752*(306*(112*(1002*(162*(540*(272*(182*(28*(2*8867310888979^2-1)^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! (4915.8475s+0.0325s) [Elapsed time: 81.95 minutes] Helper File: 547952*(135236*(20564*(1752*(306*(112*(1002*(162*(540*(272*(182*(28*(2*8867310888979^2-1)^2+1)^2-1)^2-1)^2+1)^2+1)^2+1)^2+1)^2-1)^2+1)^2-1)^2-1)^2-1
modified2020-07-07 22:30:33
created2010-08-18 18:02:31
id118034

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