# 31134000 + 3360654 + 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.

Description: 31134000 + 3360654 + 1 PRP [none] This prime has 1 user comment below. x44 : Zhou, Unknown 541056   (log10 is 541055.5028521) 2962 (digit rank is 1) 1532 yes 10/16/2017 00:22:51 CDT 10/16/2017 02:20:16 CDT 123950 Verify, TrialDiv 44.7475 (normalized score 1.4503)

Lei Zhou writes (16 Oct 2017):  (report abuse)
 This is a balanced ternary prime with only 3 non-zero digits (in balanced ternary base). p-1 = 3^1134000 + 3^360654 = Phi(4,3)*Phi(12,3)*Phi(28,3)*Phi(84,3)*Phi(73652,3)*Phi(220956,3)*Phi(515564,3)*Phi(1546692,3)*(3^360654) OpenPFGW proves that p is a Fermat and Lucas PRP. Primality testing 3^1134000+3^360654+1 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 2 Running N+1 test using discriminant 19, base 20+sqrt(19) Calling N-1 BLS with factored part 31.81% and helper 0.00% (95.42% proof) 3^1134000+3^360654+1 is Fermat and Lucas PRP! (34856.3908s+0.0187s) Then the Pari-GP code of Konyagin Pomerance method proves that p is a prime: gp >r kppm.gp gp >allocatemem(200000000) gp >N=3^1134000+3^360654+1; gp >lsm=[3^360654,2,5,73,29,16493,2857,109688713,84826040586673849]; gp >kpm(lsm,N); fraction = 318105/10^6 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 {2,5,29,73,2857,16493,109688713,84826040586673849} 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_id123950
person_id9
machineUsing: Xeon (pool) 4c+4c 3.5GHz
whatprp
notesCommand: /home/caldwell/clientpool/1/pfgw64 -t -q"3^1134000+3^360654+1" 2>&1 PFGW Version 3.7.7.64BIT.20130722.x86_Dev [GWNUM 27.11] Primality testing 3^1134000+3^360654+1 [N-1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 2 Calling Brillhart-Lehmer-Selfridge with factored part 31.81% 3^1134000+3^360654+1 is PRP! (6163.8545s+0.0224s) [Elapsed time: 1.71 hours]
modified2020-07-07 17:30:15
created2017-10-16 00:23:01
id169612

Query times: 0.0006 seconds to select prime, 0.0012 seconds to seek comments.