32095902 + 3647322 - 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:32095902 + 3647322 - 1
Verification status (*):PRP
Official Comment (*):[none]
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):x44 : Zhou, Unknown
Decimal Digits:1000000   (log10 is 999999.39200945)
Rank (*):1646 (digit rank is 2)
Entrance Rank (*):380
Currently on list? (*):short
Submitted:8/8/2018 09:57:27 CDT
Last modified:8/9/2018 17:20:20 CDT
Database id:125529
Status Flags:Verify, TrialDiv
Score (*):46.633 (normalized score 7.216)

User comments about this prime (disclaimer):

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Lei Zhou writes (8 Aug 2018):  (report abuse)
This is a balanced ternary prime with only 3 non-0 digits in balanced ternary base.
p+1 = 3^2095902 + 3^647322 = Product(Phi(m,3)), where m=(8, 24, 40, 56, 120, 168, 280, 840, 27592, 82776, 137960, 193144, 413880, 579432, 965720, 2897160)

OpenPFGW proves that p is a Fermat and Lucas PRP.
Primality testing 3^2095902+3^647322-1 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 3
Running N+1 test using discriminant 11, base 1+sqrt(11)
Calling N+1 BLS with factored part 30.89% and helper 0.00% (92.68% proof)
3^2095902+3^647322-1 is Fermat and Lucas PRP! (138312.7071s+0.0233s)

Then the Pari-GP code of Konyagin Pomerance method proves that p is a prime:
? allocatemem(16*1024*1024*1024);
? r kppm.gp;
? N=3^2095902+3^647322-1;
? lsp=[2*3^647322*241*281*337*673*1009*6481*18481*167329*298801*430697*647753*26050081*42521761*162410641*175181609*2108826721*5426131523108729*306537419965351441*369879560116990841*256392255051433268881*3353336738929580410561*257994967349862736028206417*120269035510423913774671677928007008342081*2047314589905164660182861222233071665633201];
? kpp(lsp,N)

fraction = 309138/10^6
OK -5
OK -4
OK -3
OK -2
OK -1
OK 0
OK 1
OK 2
OK 3
OK 4
OK 5

Case 1

Round of root: 0
Root OK: below the round

Other roots are complex

Case 2

Round of root:-49951...63968
Root OK: above the round

Round of root:0
Root OK: above the round

Round of root:49951...63968
Root OK: below the round

Proof completed

The prime factors used in pari KP proof as of lsp are found for the above listed Phi factors of p+1 using ECM 7.0.4.

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 -tp -q"3^2095902+3^647322-1" 2>&1 PFGW Version [GWNUM 27.11] Primality testing 3^2095902+3^647322-1 [N+1, Brillhart-Lehmer-Selfridge] Running N+1 test using discriminant 3, base 3+sqrt(3) Calling Brillhart-Lehmer-Selfridge with factored part 30.89% 3^2095902+3^647322-1 is Lucas PRP! (110619.4077s+0.0441s) [Elapsed time: 30.73 hours]
modified2020-07-07 17:30:14
created2018-08-08 10:13:02

Query times: 0.0004 seconds to select prime, 0.0005 seconds to seek comments.
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