At this site we maintain a list of the 5000 Largest Known Primes which is updated hourly. This list is the most important databases at The Prime Pages: 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:

field (help) value Description: (595310+1)2+595310 Verification status (*): Proven Official Comment: Unofficial Comments: This prime has 1 user comment below. Proof-code(s): (*): p82 : Oakes, Broadhurst, OpenPFGW Decimal Digits: 133238   (log10 is 133237.66222653) Rank (*): 6279 (digit rank is 1) Entrance Rank (*): 840 Currently on list? (*): no Submitted: 8/19/2006 11:36:02 CDT Last modified: 8/19/2006 15:11:27 CDT Removed (*): 8/11/2009 16:40:14 CDT Database id: 78328 Status Flags: none Score (*): 40.4381 (normalized score 0.8368)

User comments about this prime (disclaimer):

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David Broadhurst writes (19 Aug 2006): 

As of August 2006, (5^95310+1)^2+5^95310 = (5^95310-1)^2+5^95311 was the largest known prime of the form p=a^2+k^n=b^2+k^m with positive integers (a,b,k,m,n) and a>b. See NMBRTHRY message.

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 78328 person_id 9 machine RedHat P4 P4 what trial_divided notes Command: /home/caldwell/client/pfgw -o -f -q"(5^95310+1)^2+5^95310" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] trial factoring to 46307299 (5^95310+1)^2+5^95310 has no small factor. [Elapsed time: 524.109 seconds] modified 2007-03-03 21:00:43 created 2006-08-19 11:52:01 id 85755

field value prime_id 78328 person_id 9 machine RedHat P4 P4 what prime notes Command: /home/caldwell/client/pfgw -tc -q"(5^95310+1)^2+5^95310" 2>&1 PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4] Primality testing (5^95310+1)^2+5^95310 [N-1/N+1, Brillhart-Lehmer-Selfridge] Running N-1 test using base 2 Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(57344,20) to FFT(57344,19) Reduced from FFT(57344,19) to FFT(57344,18) Reduced from FFT(57344,18) to FFT(57344,17) Reduced from FFT(57344,17) to FFT(57344,16) 885220 bit request FFT size=(57344,16) Running N+1 test using discriminant 7, base 2+sqrt(7) Using SSE2 FFT Adjusting authentication level by 1 for PRIMALITY PROOF Reduced from FFT(57344,20) to FFT(57344,19) Reduced from FFT(57344,19) to FFT(57344,18) Reduced from FFT(57344,18) to FFT(57344,17) Reduced from FFT(57344,17) to FFT(57344,16) 885228 bit request FFT size=(57344,16) Calling N-1 BLS with factored part 50.00% and helper 0.06% (150.06% proof) (5^95310+1)^2+5^95310 is prime! (-1799.0819s+0.0300s) [Elapsed time: 11904 seconds] modified 2007-03-03 21:00:43 created 2006-08-19 11:53:03 id 85756