3^6608603 - 3^3304302 + 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:	3^6608603 - 3^3304302 + 1
Verification status (*):	Proven
Official Comment (*):	Generalized unique
Unofficial Comments:	This prime has 2 user comments below.
Proof-code(s): (*):	L5123 : Propper, Batalov, EMsieve, LLR
Decimal Digits:	3153105 (log₁₀ is 3153104.9553041)
Rank (*):	150 (digit rank is 1)
Entrance Rank (*):	107
Currently on list? (*):	yes
Submitted:	6/30/2023 17:21:33 UTC
Last modified:	3/31/2024 19:29:42 UTC
Database id:	136214
Status Flags:	none
Score (*):	50.1536 (normalized score 152.2871)

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.

Generalized Unique (archivable *)
Prime on list: yes, rank 6
Subcategory: "Generalized Unique"
(archival tag id 228750, tag last modified 2023-12-14 08:37:23)

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 (30 Jun 2023): (report abuse)

Eisenstein Mersenne Norm prime #29
This prime may also be written as: 3^6608603-3^3304302+1
Also, can be written using positive integers in cyclotomic polynomial:
Phi(3, 3^3304302 - 2)/3 or
Phi(6, 3^3304302 - 1)/3

Good reference materials for Eisenstein Mersenne Norm primes are at OEIS A066408.

Serge Batalov writes (20 Aug 2023): (report abuse)

Also, this and similar Eisenstein Mersenne Norm primes are cuban primes. This one can be represented as 3*y² + 3*y + 1 where y = 3^3304301 - 1.

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.

field value

prime_id 136214

person_id 9

machine Using: Digital Ocean Droplet

what prime

notes Command: /var/www/clientpool/1/pfgw64 -V -f -tc -q"Phi(3,-3^3304302+1)/3" 2>&1
PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11]
Primality testing Phi(3,-3^3304302+1)/3 [N-1/N+1, Brillhart-Lehmer-Selfridge]
trial

Running N-1 test using base 2
Generic modular reduction using generic reduction FMA3 FFT length 1120K, Pass1=448, Pass2=2560, clm=2 on A 10474388-bit number
Running N-1 test using base 3
Generic modular reduction using generic reduction FMA3 FFT length 1120K, Pass1=448, Pass2=2560, clm=2 on A 10474388-bit number
Running N-1 test using base 17
Generic modular reduction using generic reduction FMA3 FFT length 1120K, Pass1=448, Pass2=2560, clm=2 on A 10474388-bit number
Running N+1 test using discriminant 23, base 4+sqrt(23)
Generic modular reduction using generic reduction FMA3 FFT length 1120K, Pass1=448, Pass2=2560, clm=2 on A 10474388-bit number
Detected in MAXERR>0.45 (round off check) in Exponentiator::Iterate
Iteration: 3335823/10474395 ERROR: ROUND OFF 0.5>0.45
(Test aborted, try again using the -a1 switch)
Running N+1 test using discriminant 23, base 4+sqrt(23)
Generic modular reduction using generic reduction FMA3 FFT length 1152K, Pass1=384, Pass2=3K, clm=2 on A 10474388-bit number
Detected in MAXERR>0.45 (round off check) in Exponentiator::Iterate
Iteration: 77729/10474395 ERROR: ROUND OFF 0.5>0.45
(Test aborted, try again using the -a2 (or possibly -a0) switch)
Running N+1 test using discriminant 23, base 4+sqrt(23)
Generic modular reduction using generic reduction FMA3 FFT length 1200K, Pass1=320, Pass2=3840, clm=2 on A 10474388-bit number
Detected in MAXERR>0.45 (round off check) in Exponentiator::Iterate
Iteration: 4182708/10474395 ERROR: ROUND OFF 0.5>0.45
(Test aborted
Running N+1 test using discriminant 23, base 4+sqrt(23)
Generic modular reduction using generic reduction FMA3 FFT length 1280K, Pass1=320, Pass2=4K, clm=2 on A 10474388-bit number
Calling N-1 BLS with factored part 50.00% and helper 0.00% (150.01% proof)

Phi(3,-3^3304302+1)/3 is prime! (3258314.6117s+0.3476s)
[Elapsed time: 37.71 days]

modified 2023-08-07 10:27:16

created 2023-06-30 17:22:01

id 182045

field	value
prime_id	136214
person_id	9
machine	Using: Digital Ocean Droplet
what	prime
notes	Command: /var/www/clientpool/1/pfgw64 -V -f -tc -q"Phi(3,-3^3304302+1)/3" 2>&1 PFGW Version 4.0.4.64BIT.20221214.x86_Dev [GWNUM 30.11] Primality testing Phi(3,-3^3304302+1)/3 [N-1/N+1, Brillhart-Lehmer-Selfridge] trial Running N-1 test using base 2 Generic modular reduction using generic reduction FMA3 FFT length 1120K, Pass1=448, Pass2=2560, clm=2 on A 10474388-bit number Running N-1 test using base 3 Generic modular reduction using generic reduction FMA3 FFT length 1120K, Pass1=448, Pass2=2560, clm=2 on A 10474388-bit number Running N-1 test using base 17 Generic modular reduction using generic reduction FMA3 FFT length 1120K, Pass1=448, Pass2=2560, clm=2 on A 10474388-bit number Running N+1 test using discriminant 23, base 4+sqrt(23) Generic modular reduction using generic reduction FMA3 FFT length 1120K, Pass1=448, Pass2=2560, clm=2 on A 10474388-bit number Detected in MAXERR>0.45 (round off check) in Exponentiator::Iterate Iteration: 3335823/10474395 ERROR: ROUND OFF 0.5>0.45 (Test aborted, try again using the -a1 switch) Running N+1 test using discriminant 23, base 4+sqrt(23) Generic modular reduction using generic reduction FMA3 FFT length 1152K, Pass1=384, Pass2=3K, clm=2 on A 10474388-bit number Detected in MAXERR>0.45 (round off check) in Exponentiator::Iterate Iteration: 77729/10474395 ERROR: ROUND OFF 0.5>0.45 (Test aborted, try again using the -a2 (or possibly -a0) switch) Running N+1 test using discriminant 23, base 4+sqrt(23) Generic modular reduction using generic reduction FMA3 FFT length 1200K, Pass1=320, Pass2=3840, clm=2 on A 10474388-bit number Detected in MAXERR>0.45 (round off check) in Exponentiator::Iterate Iteration: 4182708/10474395 ERROR: ROUND OFF 0.5>0.45 (Test aborted Running N+1 test using discriminant 23, base 4+sqrt(23) Generic modular reduction using generic reduction FMA3 FFT length 1280K, Pass1=320, Pass2=4K, clm=2 on A 10474388-bit number Calling N-1 BLS with factored part 50.00% and helper 0.00% (150.01% proof) Phi(3,-3^3304302+1)/3 is prime! (3258314.6117s+0.3476s) [Elapsed time: 37.71 days]
modified	2023-08-07 10:27:16
created	2023-06-30 17:22:01
id	182045

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

36608603 - 33304302 + 1

This prime's information:

Archival tags:

User comments about this prime (disclaimer):

Verification data:

3^6608603 - 3^3304302 + 1