2 · 310852677 + 1

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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:2 · 310852677 + 1
Verification status (*):Proven
Official Comment (*):Divides phi
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):L4965 : Propper, LLR
Decimal Digits:5178044   (log10 is 5178043.1683372)
Rank (*):43 (digit rank is 1)
Entrance Rank (*):30
Currently on list? (*):yes
Submitted:1/8/2023 17:42:38 UTC
Last modified:5/20/2023 20:59:19 UTC
Database id:134762
Status Flags:none
Score (*):51.6727 (normalized score 693.9384)

Archival tags:

There are certain forms classed as archivable: these prime may (at times) remain on this list even if they do not make the Top 5000 proper.  Such primes are tracked with archival tags.

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.

Ryan Propper writes (10 Jan 2023):  (report abuse) 
Starting test if 2*3^10852677+1 divides Phi(3^10852677,2)
Using all-complex AVX-512 FFT length 960K, Pass1=960, Pass2=1K, clm=2, 16 threads, a = 2
2*3^10852677+1 Divides 2^3^10852677-1 Time : 10050.323 sec.

Starting test if 2*3^10852677+1 divides Phi(3^10852676,2)
Using all-complex AVX-512 FFT length 960K, Pass1=960, Pass2=1K, clm=2, 16 threads, a = 2
2*3^10852677+1 Divides 2^3^10852676-1 Time : 10039.650 sec.

Starting test if 2*3^10852677+1 divides Phi(3^10852675,2)
Using all-complex AVX-512 FFT length 960K, Pass1=960, Pass2=1K, clm=2, 16 threads, a = 2
2*3^10852677+1 Divides 2^3^10852675-1 Time : 9945.409 sec.

Starting test if 2*3^10852677+1 divides Phi(3^10852674,2)
Using all-complex AVX-512 FFT length 960K, Pass1=960, Pass2=1K, clm=2, 16 threads, a = 2
2*3^10852677+1 Divides 2^3^10852674-1 Time : 18489.739 sec.

Starting test if 2*3^10852677+1 divides Phi(3^10852673,2)
Using all-complex AVX-512 FFT length 960K, Pass1=960, Pass2=1K, clm=2, 16 threads, a = 2
2*3^10852677+1 does not divide 2^3^10852673-1 Time : 23416.250 sec. 

Conclusion: 2*3^10852677+1 Divides Phi(3^10852674,2)

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_id134762
person_id9
machineUsing: Dual Intel Xeon Gold 5222 CPUs 3.8GHz
whatprime
notesCommand: /home/caldwell/clientpool/1/pfgw64 -t -q"2*3^10852677+1" 2>&1
PFGW Version 4.0.1.64BIT.20191203.x86_Dev [GWNUM 29.8]
Primality testing 2*3^10852677+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
Calling Brillhart-Lehmer-Selfridge with factored part 100.00%


2*3^10852677+1 is prime! (49194.5292s+0.0579s)
[Elapsed time: 13.66 hours]
modified2023-01-09 07:27:56
created2023-01-08 17:48:02
id180542

Query times: 0.0003 seconds to select prime, 0.0004 seconds to seek comments.
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