(2²⁸¹⁶²¹ + 1)² - 2

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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:	(2²⁸¹⁶²¹ + 1)² - 2
Verification status (*):	Proven
Official Comment (*):	[none]
Proof-code(s): (*):	p89 : Emmanuel, OpenPFGW
Decimal Digits:	169553 (log₁₀ is 169552.73681777)
Rank (*):	38973 (digit rank is 1)
Entrance Rank (*):	347
Currently on list? (*):	no
Submitted:	10/9/2005 13:15:23 UTC
Last modified:	3/11/2023 15:54:10 UTC
Removed (*):	10/26/2010 16:01:27 UTC
Database id:	75878
Status Flags:	none
Score (*):	41.1801 (normalized score 0.0186)

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 75878

person_id 9

machine Linux P4 2.8GHz

what prime

notes Command: /home/caldwell/client/pfgw -f -tc -q"(2^281621+1)^2-2" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing (2^281621+1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge] trial factoring to 59879187 Running N-1 test using base 3 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(65536,20) to FFT(65536,19) Reduced from FFT(65536,19) to FFT(65536,18) 1126494 bit request FFT size=(65536,18) Running N+1 test using discriminant 7, base 6+sqrt(7) Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(65536,20) to FFT(65536,19) Reduced from FFT(65536,19) to FFT(65536,18) 1126502 bit request FFT size=(65536,18) Running N+1 test using discriminant 7, base 9+sqrt(7) Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(65536,20) to FFT(65536,19) Reduced from FFT(65536,19) to FFT(65536,18) 1126502 bit request FFT size=(65536,18) Calling N+1 BLS with factored part 50.01% and helper 0.00% (150.02% proof) (2^281621+1)^2-2 is prime! (-946.5011s+0.0200s) [Elapsed time: 61891 seconds]

modified 2020-07-07 22:30:43

created 2005-10-09 15:08:02

id 81072

field	value
prime_id	75878
person_id	9
machine	Linux P4 2.8GHz
what	prime
notes	Command: /home/caldwell/client/pfgw -f -tc -q"(2^281621+1)^2-2" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing (2^281621+1)^2-2 [N-1/N+1, Brillhart-Lehmer-Selfridge] trial factoring to 59879187 Running N-1 test using base 3 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(65536,20) to FFT(65536,19) Reduced from FFT(65536,19) to FFT(65536,18) 1126494 bit request FFT size=(65536,18) Running N+1 test using discriminant 7, base 6+sqrt(7) Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(65536,20) to FFT(65536,19) Reduced from FFT(65536,19) to FFT(65536,18) 1126502 bit request FFT size=(65536,18) Running N+1 test using discriminant 7, base 9+sqrt(7) Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(65536,20) to FFT(65536,19) Reduced from FFT(65536,19) to FFT(65536,18) 1126502 bit request FFT size=(65536,18) Calling N+1 BLS with factored part 50.01% and helper 0.00% (150.02% proof) (2^281621+1)^2-2 is prime! (-946.5011s+0.0200s) [Elapsed time: 61891 seconds]
modified	2020-07-07 22:30:43
created	2005-10-09 15:08:02
id	81072

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

(2281621 + 1)2 - 2

This prime's information:

Verification data:

(2²⁸¹⁶²¹ + 1)² - 2