2 · 47346759 + 1
(Another of the Prime Pages' resources)
The Largest Known Primes Icon
  View this page in:   language help
 

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:2 · 47346759 + 1
Verification status (*):Proven
Official Comment:Divides Phi(47^346759,2)
Unofficial Comments:This prime has 1 user comment below.
Proof-code(s): (*):g424 : Broadhurst, NewPGen, OpenPFGW, Proth.exe
Decimal Digits:579816   (log10 is 579815.28214993)
Rank (*):1293 (digit rank is 1)
Entrance Rank (*):87
Currently on list? (*):short
Submitted:1/26/2011 22:04:00 CDT
Last modified:1/27/2011 20:21:02 CDT
Database id:97765
Status Flags:none
Score (*):44.9599 (normalized score 3.3854)

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.
Divides Phi (archivable *)
Prime on list: yes, rank 8
Subcategory: "Divides Phi"
(archival tag id 213110, tag last modified 2015-03-04 06:50:26)

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.

David Broadhurst writes (11 Sep 2014): 
OpenPFGW found that 2*47^n+1 is prime for n=346759 and that the only smaller positive integer yielding a prime is n=175. Proth.exe found that the order of 2, modulo p=2*47^346759+1, is a prime power, namely (p-1)/2. With 579816 decimal digits, p more than doubled the number of digits of the previous record for this form of prime, which was 2*827^98511+1, with 287407 digits. I thank Wilfrid Keller for encouragement, over the last 7 years, to find a large prime of this form.

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_id97765
person_id9
machineDitto P4 P4
whattrial_divided
notesCommand: /home/ditto/client/TrialDiv/TrialDiv -q 2 47 346759 1 2>&1
[Elapsed time: 9.940 seconds]
modified2011-12-27 16:48:35
created2011-01-26 22:05:02
id124919

fieldvalue
prime_id97765
person_id9
machineRedHat P4 P4
whatprime
notesCommand: /home/caldwell/client/pfgw -t -q"2*47^346759+1" 2>&1
PFGW Version 20031027.x86_Dev (Beta 'caveat utilitor') [FFT v22.13 w/P4]
Primality testing 2*47^346759+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 5
Calling Brillhart-Lehmer-Selfridge with factored part 100.00%
2*47^346759+1 is prime! (-849.1840s+0.2400s)
[Elapsed time: 21.90 hours]
modified2011-03-25 10:27:31
created2011-01-26 22:23:01
id124920

Query times: 0.0005 seconds to select prime, 0.0007 seconds to seek comments.