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82000-digit prime with all digits prime

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David Broadhurst     Message 1 of 2  Oct 20, 2003
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N=(10^40950+1)*(10^20055+1)*(10^10374+1)*
(10^4955+1)*(10^2507+1)*(10^1261+1)*
(3*R(1898)+555531001*10^940-R(958))+1
is a prime all of whose 82000 decimal digits are prime.

The Ansatz was selected after detailed scrutiny of the
the database of factors of 10^n+1 maintained by Greg Childers.

Phil Carmody sieved a range of candidates to a depth of 0.46T.

OpenPFGW found that N is probably prime.

VFYPR proved the primality of 238 factors of N-1
with up to 237 digits.

Primo proved the primality of 9 larger factors of N-1, namely:

//p713=
Phi(2*955,10)/805712491/8216075917481/128275956445522068558281771

//p952=
Phi(2*991,10)/154326449/1040954329/769150565783664091847

//p1148=
Phi(2*1261,10)/22699

//p1416=
gcd(Phi(2*5850,10),10^(4*585)+5*10^(3*585)+7*10^(2*585)+5*10^585+1\
+10^((585+1)/2)*(10^(3*585)+2*10^(2*585)+2*10^585+1))\
/13667940001/170791741656901

//p1440=
gcd(Phi(2*4550,10),10^(4*455)+5*10^(3*455)+7*10^(2*455)+5*10^455+1\
+10^((455+1)/2)*(10^(3*455)+2*10^(2*455)+2*10^455+1))

//p2367=
Phi(2*2507,10)/60169/65183

//p2567=
Phi(2*3458,10)/20749/2074801/16204189/65321621

//p3951=
Phi(2*4955,10)/6612070921

//p4560=
Phi(2*6685,10) 

OpenPFGW performed the BLS tests:

> [N-1, Brillhart-Lehmer-Selfridge]
> Running N-1 test using base 3
> Running N-1 test using base 5
> Calling Brillhart-Lehmer-Selfridge with factored part 30.18%
> (10^40950+1)*(10^20055+1)*(10^10374+1)*
> (10^4955+1)*(10^2507+1)*(10^1261+1)*
> (3*R(1898)+555531001*10^940-R(958))+1
> is PRP! (22606.887000 seconds)

Pari-GP completed the Konyagin-Pomerance proof.

David Broadhurst
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David Broadhurst     Message 2 of 2  Oct 25, 2003
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> //p3951=
> Phi(2*4955,10)/6612070921

took merely 370 hours to prove, at 1.0 Ghz:

> [PRIMO - Primality Certificate]
> Version=2.2.0 beta 1
> WebSite=http://www.ellipsa.net/
<snip>
> Initialization=46.84s
> 1stPhase=305h 7mn 29s
> 2ndPhase=65h 5mn 34s
> Total=370h 13mn 50s
<snip>
> BinarySize=13123

My old guestimator suggested
(13123/3000)^4.5=765 GHz-hours.

So this was twice as fast as most of my
experience with versions previous to 2.2.0 beta 1.

David
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