31681130 + 3445781 + 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:31681130 + 3445781 + 1
Verification status (*):PRP
Official Comment (*):[none]
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):CH9 : Zhou, OpenPFGW, CHG
Decimal Digits:802103   (log10 is 802102.85494687)
Rank (*):2959 (digit rank is 1)
Entrance Rank (*):2337
Currently on list? (*):short
Submitted:9/1/2022 22:58:33 UTC
Last modified:5/20/2023 20:59:19 UTC
Database id:134345
Status Flags:Verify
Score (*):45.9563 (normalized score 3.2144)

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 (5 Sep 2022):  (report abuse)
This is a balanced ternary prime with only 3 non-0 digits in balanced ternary base.
p-1 = 3^1681130+3^445781 = 3^445781 * Product(Phi(m,3)), where m = (2, 6, 18, 634,866, 1902, 2598, 5706, 7794, 274522, 823566, 2470698).
OpenPFGW proves that p is a Fermat and Lucas PRP in about 1 day.
Then the Pari-GP code of CHG.GP proves that p is a prime.
: $ gp
Input file is:
Certificate file is:
... Search for factors congruent to 1.
Running CHG with h = 16, u = 7. Right endpoint has 161883 digits. Done!
Time elapsed: 810732137ms.
Running CHG with h = 7, u = 2. Right endpoint has 57119 digits. Done! Time elapsed: 4457878ms. A certificate has been saved to the file: cp_1681130_+3_445781_c1_+1.out
Running David Broadhurst's verifier on the saved certificate...
Testing a PRP called "cp_1681130_+3_445781_c1_+1.in".
Validated in 116 sec. Congratulations! n is prime! Goodbye!
The above proof was generated in about 33 days using a Intel(R) Xeon(R) Silver 4215 CPU @ 2.50GHz as a single thread.
This is the most time consuming prime proof using CHG method up to date (Sept. 1st, 2022).
A full certificate can be found 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.
machineUsing: Dual Intel Xeon Gold 5222 CPUs 3.8GHz
notesCommand: /home/caldwell/clientpool/1/pfgw64 -t -q"3^1681130+3^445781+1" 2>&1
PFGW Version [GWNUM 29.8]
Primality testing 3^1681130+3^445781+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
Calling Brillhart-Lehmer-Selfridge with factored part 26.52%

3^1681130+3^445781+1 is PRP! (7156.9477s+0.0079s)
[Elapsed time: 1.99 hours]
modified2022-09-02 01:03:18
created2022-09-01 23:04:01

Query times: 0.0002 seconds to select prime, 0.0004 seconds to seek comments.
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