1 - V( - 3, - 3, 1307101) - 31307101

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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:1 - V( - 3, - 3, 1307101) - 31307101
Verification status (*):Proven
Official Comment (*):[none]
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):p437 : Propper, Batalov, EMsieve, PIES, OpenPFGW
Decimal Digits:756533   (log10 is 756532.75120575)
Rank (*):4081 (digit rank is 1)
Entrance Rank (*):3987
Currently on list? (*):yes
Submitted:10/29/2024 05:26:10 UTC
Last modified:10/30/2024 10:37:16 UTC
Database id:138685
Status Flags:none
Score (*):45.7767 (normalized score 1.8214)

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 (29 Oct 2024):  (report abuse)
Primes of this form are "cousins" of Mersenne, Fermat, Gaussian-Mersenne and Eisenstein-Mersenne norm primes. See Mike Oakes' observations from 2002. D.Broadhurst called those extensions "Oakes primes".

The reason the primes of this form with b = -2 and b = -3 (like this one) are provable with N-1 BLS proof is that N-1 is > 33% factored:
-lucasV(-2,-2,n) = 2^k * A001834(k-1) for odd n = 2k - 1, where A001834(i) = lucasU(4,1,i)+lucasU(4,1,i+1), and
-lucasV(-3,-3,n) = 3^k * A030221(k-1) for odd n = 2k - 1, where A030221(i) = (lucasV(5,1,i+1)-lucasV(5,1,i))/3.

The A() cofactors are somewhat connected to Lehmer numbers.

Furthermore, primes in all these series are only possible with n prime.

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_id138685
person_id9
machineUsing: Digital Ocean Droplet
whatprime
notesCommand: /var/www/clientpool/1/pfgw64 -V -f -tc -q"1-lucasV(-3,-3,1307101)-3^1307101" >command_output 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing 1-lucasV(-3,-3,1307101)-3^1307101 [N-1/N+1, Brillhart-Lehmer-Selfridge]
trial


Running N-1 test using base 11
Generic modular reduction using generic reduction AVX-512 FFT length 280K, Pass1=128, Pass2=2240, clm=2 on A 2513148-bit number
Running N-1 test using base 13
Generic modular reduction using generic reduction AVX-512 FFT length 280K, Pass1=128, Pass2=2240, clm=2 on A 2513148-bit number
Running N+1 test using discriminant 19, base 9+sqrt(19)
Generic modular reduction using generic reduction AVX-512 FFT length 280K, Pass1=128, Pass2=2240, clm=2 on A 2513148-bit number
Calling N-1 BLS with factored part 41.22% and helper 0.00% (123.66% proof)


1-lucasV(-3,-3,1307101)-3^1307101 is prime! (60261.8269s+0.0384s)
[Elapsed time: 16.74 hours]
modified2024-10-30 09:38:49
created2024-10-29 16:54:27
id184528

Query times: 0.0002 seconds to select prime, 0.0003 seconds to seek comments.
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