145950 · ((46953 · (278822 · ((33410 · (6470 · ((884 · (466 · ((78 · (5159780352 · 53125 + 7))3 + 1) + 1))3 + 1) + 1))3 + 1) + 1))3 + 1) + 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:145950 · ((46953 · (278822 · ((33410 · (6470 · ((884 · (466 · ((78 · (5159780352 · 53125 + 7))3 + 1) + 1))3 + 1) + 1))3 + 1) + 1))3 + 1) + 1
Verification status (*):Proven
Official Comment (*):[none]
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):p298 : Zhou, Primo, OpenPFGW
Decimal Digits:178129   (log10 is 178128.89397171)
Rank (*):39703 (digit rank is 1)
Entrance Rank (*):4785
Currently on list? (*):no
Submitted:11/21/2010 22:34:38 UTC
Last modified:3/11/2023 15:54:10 UTC
Removed (*):12/16/2010 15:21:09 UTC
Database id:96540
Status Flags:none
Score (*):41.3319 (normalized score 0.0144)

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)
The kernel "12^9*5^5^5+7" is proven by Primo
This prime is then proven using pfgw in the following sequence:
p0=12^9*5^5^5+7;
p1=(a0*p0)^2-a0*p0+1 with a0=78, using p0 as helper;
p2=b0*((a0*p0)^3+1)+1 with b0=466, using p1 as helper;
p3=(a2*p2)^2-a2*p2+1 with a2=884, using p2 as helper;
p4=b2*((a2*p2)^3+1)+1 with b2=6470, using p2 as helper;
p5=(a4*p4)^2-a4*p4+1 with a4=33410, using p4 as helper;
p6=b4*((a4*p4)^3+1)+1 with b4=278822, using p4 as helper;
p7=(a6*p6)^2-a6*p6+1 with a6=46953, using p6 as helper;
p8=b6*((a6*p6)^3+1)+1 with b6=145950, using p6 as helper. The detail is HERE.
Proof code:
#!/bin/sh
./pfgw -l"pmtup_5.4.2.cert" -t -h"p_00" ph_01
./pfgw -l"pmtup_5.4.2.cert" -t -h"ph_01" p_01
./pfgw -l"pmtup_5.4.2.cert" -t -h"p_01" ph_02
./pfgw -l"pmtup_5.4.2.cert" -t -h"ph_02" p_02
./pfgw -l"pmtup_5.4.2.cert" -t -h"p_02" ph_03
./pfgw -l"pmtup_5.4.2.cert" -t -h"ph_03" p_03
./pfgw -l"pmtup_5.4.2.cert" -t -h"p_03" ph_04
./pfgw -l"pmtup_5.4.2.cert" -t -h"ph_04" p_04

Helpers:
p_00: 12^9*5^3125+7
ph_01: (78*(12^9*5^3125+7)-1)*(78*(12^9*5^3125+7))+1
p_01: 466*((78*(12^9*5^3125+7))^3+1)+1
ph_02: (884*(466*((78*(12^9*5^3125+7))^3+1)+1)-1)*(884*(466*((78*(12^9*5^3125+7))^3+1)+1))+1
p_02: 6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1
ph_03: (33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1)-1)*(33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))+1
p_03: 278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1
ph_04: (46953*(278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1)-1)*(46953*(278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))+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_id96540
person_id9
machineDitto P4 P4
whattrial_divided
notesPFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] 145950*((46953*(278822*((.........^3+1)+1))^3+1)+1 1/1 trial factoring to 63112283 145950*((46953*(278822*((33410*(6470*((884*(466*((78*(5159780352*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))^3+1)+1 has no small factor. [Elapsed time: 826.630 seconds]
modified2020-07-07 22:30:33
created2010-11-21 22:35:01
id122466

fieldvalue
prime_id96540
person_id9
machineRedHat Virtual STEM Server
whatprime
notesPFGW Version 3.3.4.20100405.x86_Stable [GWNUM 25.14] Primality testing 145950*((46953*(278822*((33410*(6470*((884*(466*((78*(5159780352*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))^3+1)+1 [N-1, Brillhart-Lehmer-Selfridge] Reading factors from helper file helper Running N-1 test using base 11 Calling Brillhart-Lehmer-Selfridge with factored part 66.66% 145950*((46953*(278822*((33410*(6470*((884*(466*((78*(5159780352*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))^3+1)+1 is prime! (3401.0967s+0.0465s) [Elapsed time: 56.87 minutes] Helper File: (78*(12^9*5^3125+7)-1)*(78*(12^9*5^3125+7))+1 (884*(466*((78*(12^9*5^3125+7))^3+1)+1)-1)*(884*(466*((78*(12^9*5^3125+7))^3+1)+1))+1 (33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1)-1)*(33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))+1 (46953*(278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1)-1)*(46953*(278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1))+1 12^9*5^3125+7 466*((78*(12^9*5^3125+7))^3+1)+1 6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1 278822*((33410*(6470*((884*(466*((78*(12^9*5^3125+7))^3+1)+1))^3+1)+1))^3+1)+1
modified2020-07-07 22:30:33
created2010-11-28 20:28:00
id122627

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