32120580 - 3623816 - 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:32120580 - 3623816 - 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:1011774   (log10 is 1011773.7903334)
Rank (*):2449 (digit rank is 1)
Entrance Rank (*):414
Currently on list? (*):short
Submitted:5/12/2019 13:20:20 UTC
Last modified:5/20/2023 20:59:19 UTC
Database id:126454
Status Flags:Verify
Score (*):46.6689 (normalized score 5.1332)

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 (12 May 2019):  (report abuse)
This is a balanced ternary prime with 3 non-zero digits (in balanced ternary base).

p+1=3^2120580-3^623816=Phi(1,3)*Phi(2,3)*Phi(4,3)*Phi(383,3)*Phi(766,3)*Phi(977,3)*Phi(1532,3)*Phi(1954,3)*Phi(3908,3)*Phi(374191,3)*Phi(748382,3)*Phi(1496764,3) which was found with a list of small factors in the proof pack here.

Using the above small factors as help, OpenPFGW provides that it is a Fermat and Lucas PRP:
$ ./pfgw -tc -k -h"cp_2120580_-3_623816_-1.helper" cp_2120580_-3_623816_-1
Primality testing 3^2120580-3^623816-1 [N-1/N+1, Brillhart-Lehmer-Selfridge]
Reading factors from helper file cp_2120580_-3_623816_-1.helper
Running N-1 test using base 41
Running N+1 test using discriminant 47, base 1+sqrt(47)
3^2120580-3^623816-1 is Fermat and Lucas PRP! (141118.4747s+0.0156s)

Using F = 2 and G = the product of the above listed small factors, CHG pari script proved that this is a prime number. (full CHG output is too long to be posted here. Verification is the following)
The result is certified by David Broadhurst's verifier chgcertd.gp.
Testing a PRP called "cp_2120580_-3_623816_-1.in".

Pol[1, 1] with [h, u]=[4, 1] has ratio=2.796269296969836082 E-179332 at X, ratio=1.9778997813013111843 E-125427 at Y, witness=2.
Pol[2, 1] with [h, u]=[6, 2] has ratio=2.4698351139240039616 E-53343 at X, ratio=3.450462696097655256 E-63276 at Y, witness=2.

Validated in 8 sec.

The certificate is available 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: Xeon 4c+4c 3.5GHz
notesCommand: /home/caldwell/client/pfgw/pfgw64 -tp -q"3^2120580-3^623816-1" 2>&1 PFGW Version [GWNUM 27.11] Primality testing 3^2120580-3^623816-1 [N+1, Brillhart-Lehmer-Selfridge] Running N+1 test using discriminant 41, base 1+sqrt(41) Calling Brillhart-Lehmer-Selfridge with factored part 29.42% 3^2120580-3^623816-1 is Lucas PRP! (101371.4113s+0.0429s) [Elapsed time: 28.16 hours]
modified2020-07-07 22:30:13
created2019-05-12 13:21:01

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