3850500 + 3248918 - 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 databases at The Prime Pages: 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:

field (help)value
Description:3850500 + 3248918 - 1
Verification status (*):PRP
Official Comment:
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):CH9 : Zhou, OpenPFGW, CHG
Decimal Digits:405792   (log10 is 405791.627139073)
Rank (*):4305 (digit rank is 1)
Entrance Rank (*):3201
Currently on list? (*):yes
Submitted:7/16/2016 09:27:25 CDT
Last modified:7/16/2016 12:50:36 CDT
Database id:121904
Status Flags:Verify, TrialDiv
Score (*):43.8637 (normalized score 1.1121)

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 (16 Jul 2016): 
This is a balanced ternary prime with only 3 non-zero digits (in balanced ternary base).

p+1 = 3^850500 + 3^248918 = Phi(4,3)*Phi(244,3)*Phi(19724,3)*Phi(1203164,3) which has the following small factors: 2, 3^248918, 5, 98621, 180317, 1518749, 1538473, 262199473, 14033606981014657, 39504363995133913, 125904201746877738245401, 865923475887669700104067517
Using the above small factors as help, OpenPFGW provides 87.89% proof:
Primality testing 3^850500+3^248918-1 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Primality testing 3^850500+3^248918-1 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Running N+1 test using discriminant 17, base 8+sqrt(17)
Calling N+1 BLS with factored part 29.30% and helper 0.00% (87.89% proof)
3^850500+3^248918-1 is Fermat and Lucas PRP! (22180.7503s+0.0095s)

Using F = 2 and G = the product of the above listed small factors, CHG pari script proved that this is a prime number.
(full CHG output is too long to be posted here. Verification is the following)
The result is certified by David Broadhurst's verifier chgcertd.gp.
Testing a PRP called "cp_850500_+3_248918_-1.in".

Pol[1, 1] with [h, u]=[4, 1] has ratio=6.850021549422434505 E-69380 at X, ratio=9.193825434597816829 E-81355 at Y, witness=2.
Pol[2, 1] with [h, u]=[5, 2] has ratio=1.0279009322513632653 E-81354 at X, ratio=1.0138544926424912111 E-40677 at Y, witness=2.
Pol[3, 1] with [h, u]=[6, 2] has ratio=3.046034677385834765 E-28588 at X, ratio=3.578658306755167131 E-20454 at Y, witness=5.


Validated in 6 sec.

The certificate is available at here.

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_id121904
person_id9
machineUsing: Xeon 4c+4c 3.5GHz
whatprp
notesCommand: /home/caldwell/client/pfgw/pfgw64 -tp -q"3^850500+3^248918-1" 2>&1
PFGW Version 3.7.7.64BIT.20130722.x86_Dev [GWNUM 27.11]
Primality testing 3^850500+3^248918-1 [N+1, Brillhart-Lehmer-Selfridge]
Running N+1 test using discriminant 3, base 3+sqrt(3)
Calling Brillhart-Lehmer-Selfridge with factored part 29.27%


3^850500+3^248918-1 is Lucas PRP! (11378.9385s+0.0152s)
[Elapsed time: 3.16 hours]
modified2017-01-27 05:57:11
created2016-07-16 09:31:01
id167543

Query times: 0.0005 seconds to select prime, 0.0007 seconds to seek comments.