31170000 + 3364398 + 1

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:31170000 + 3364398 + 1
Verification status (*):PRP
Official Comment (*):[none]
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):x44 : Zhou, Unknown
Decimal Digits:558232   (log10 is 558231.868022)
Rank (*):4371 (digit rank is 1)
Entrance Rank (*):1238
Currently on list? (*):yes
Submitted:1/7/2017 05:26:30 UTC
Last modified:3/11/2023 15:54:10 UTC
Database id:122699
Status Flags:Verify, TrialDiv
Score (*):44.8434 (normalized score 1.0774)

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 (7 Jan 2017):  (report abuse)
This is a balanced ternary prime with only 3 non-zero digits (in balanced ternary base).
p-1 = 3^1170000+3^364398 = Phi(4,3)*Phi(12,3)*Phi(28,3)*Phi(84,3)*Phi(76724,3)*Phi(230172,3)*Phi(537068,3)*Phi(1611204,3)*(3^364398)
OpenPFGW proves that p is a Fermat and Lucas PRP.
Primality testing 3^1170000+3^364398+1 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Running N+1 test using discriminant 19, base 9+sqrt(19)
Calling N-1 BLS with factored part 31.15% and helper 0.00% (93.46% proof)
3^1170000+3^364398+1 is Fermat and Lucas PRP! (34157.1809s+0.0273s)

Then the Pari-GP code of Konyagin Pomerance method proves that p is a prime:
gp >r kppm.gp
gp >allocatemem(80000000)
gp >N=3^1170000+3^364398+1
gp >lsm=[3^364398,2,5,29,73,2857,16493,109688713,182364278189,1008674971106237,16331748023034276941]
gp >kpm(lsm,N)

fraction = 311569/10^6
OK 0
OK 1
OK 2
OK 3
OK 4
OK 5

Round of root:
Root OK: above the round

Other roots are complex

Proof completed

where prime factor set {2,5,29,73,2857,16493,109688713,182364278189,1008674971106237,16331748023034276941} of p-1 are found from small Phi factors of p-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.
machineUsing: Xeon (pool) 4c+4c 3.5GHz
notesCommand: /home/caldwell/clientpool/1/pfgw64 -t -q"3^1170000+3^364398+1" 2>&1 PFGW Version [GWNUM 27.11] Primality testing 3^1170000+3^364398+1 [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 2 Calling Brillhart-Lehmer-Selfridge with factored part 31.15% 3^1170000+3^364398+1 is PRP! (6505.3525s+0.0214s) [Elapsed time: 1.81 hours]
modified2020-07-07 22:30:16
created2017-01-07 05:33:01

Query times: 0.0006 seconds to select prime, 0.0005 seconds to seek comments.
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