# 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 PRP [none] This prime has 1 user comment below. x44 : Zhou, Unknown 558232   (log10 is 558231.868022) 4366 (digit rank is 1) 1238 yes 1/7/2017 05:26:30 UTC 3/11/2023 15:54:10 UTC 122699 Verify, TrialDiv 44.8434 (normalized score 1.0784)

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: 0 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.
fieldvalue
prime_id122699
person_id9
machineUsing: Xeon (pool) 4c+4c 3.5GHz
whatprp
notesCommand: /home/caldwell/clientpool/1/pfgw64 -t -q"3^1170000+3^364398+1" 2>&1 PFGW Version 3.7.7.64BIT.20130722.x86_Dev [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
id168342

Query times: 0.0005 seconds to select prime, 0.0011 seconds to seek comments.