1024 · 31877301 + 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:1024 · 31877301 + 1
Verification status (*):Proven
Official Comment (*):[none]
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):p378 : Batalov, Srsieve, CRUS, LLR, OpenPFGW
Decimal Digits:895704   (log10 is 895703.21890644)
Rank (*):4225 (digit rank is 1)
Entrance Rank (*):160
Currently on list? (*):yes
Submitted:12/15/2014 05:05:01 UTC
Last modified:8/25/2025 02:22:42 UTC
Database id:118946
Status Flags:none
Score (*):46.295 (normalized score 1.7759)

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.

Serge Batalov writes (16 Dec 2014):  (report abuse)
Short summary of findings:
1) Currently (as of Dec/15/2014), there is a mixed bag of this number's character results. It is unclear where the software bug precisely is.
2) this number is a 2-, 3-, 5-, 7-, 11-, 13-, 17-, 19-PRP by Prime95 (mprime 27.9, 28.5 were tested)
3) this number is reported prime by N-1 test with non-AVX (two different FFT sizes) PFGW (with pre-AVX version of PFGW or latest PFGW on non-AVX hardware)
4) this number is reported prime by N-1 test with LLR 3.8.13 on non-AVX hardware, using two different FFT sizes: 256K, 288K, and with FermatBase=5, 7, 11, 19
but
5) this number is reported composite by AVX-enabled versions of PFGW and LLR (however, it is a 3- and 17-SPRP in AVX-FFT LLR)

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_id118946
person_id9
machineUsing: Digital Ocean Droplet
whatprime
notesCommand: /var/www/clientpool/1/pfgw64 -V -f -t -q"1024*3^1877301+1" >command_output 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing 1024*3^1877301+1 [N-1, Brillhart-Lehmer-Selfridge]
trial


Running N-1 test using base 5
Special modular reduction using all-complex FMA3 FFT length 240K, Pass1=1280, Pass2=192, clm=2 on 1024*3^1877301+1
Calling Brillhart-Lehmer-Selfridge with factored part 100.00%


1024*3^1877301+1 is prime! (5468.9345s+0.0095s)
[Elapsed time: 1.52 hours]
modified2025-08-25 02:22:42
created2025-08-25 00:51:29
id187114

Query times: 0.0004 seconds to select prime, 0.0006 seconds to seek comments.
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