## Definitive answer to post 10359 19 Dec 2002

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• Hi All, Long long ago 19 Dec 2002 (post 10359) Sebastián Martín Ruiz asked Let F(n)=P(n)!-P(n)!!+2 P(n) n-th prime F(1)=2 Prime F(2)=5 Prime F(3)=107 Prime
Message 1 of 1 , Aug 30, 2004
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Hi All,
Long long ago 19 Dec 2002 (post 10359) Sebastián Martín Ruiz asked

Let F(n)=P(n)!-P(n)!!+2

P(n) n-th prime

F(1)=2 Prime
F(2)=5 Prime
F(3)=107 Prime
F(4)=4937 Prime

Find another prime number in this secuence

to which I replied (without proof as I could only show that it was
Fermat and Lucas PRP)
(1667!)-(1667!2)+2
However as it has 4650 digits I thought it should be provable.
David Broadhurst pointed me to Marcel Martin's Primo (a marvellous
program).

I started primo 2.0.0 on 1667!-1667!2+2 and left it for what I
thought
would be 6-12 months.
It has finished after "only" 1340h 11mn 39s
the result being

Version=2.0.0 - beta 3
ID=B285E020B5C1A
Created=01/15/2003 09:31:38 AM

[Common]
Path=C:\Program Files\Primo200\f1667\
Selected=1
Processed=1
Certified=0
Candidate #1=Failed, 1340h 11mn 39s

[Candidate #1]
Input=Primo-B284302D531FB-001.tmp
Output=Primo-B284302D531FB-001.tmp
Status=Failed, impossible to certify current N

Not! what I was hoping for.

Changes to Primo made it worthwhile trying again
This time Primo sucessfully proved 1667!-1667!2+2 Prime

[PRIMO - Primality Certificate]
Version=2.2.0 beta 3
WebSite=http://www.ellipsa.net/
Format=3
ID=B2A0002ACB343
Created=03/08/2004 12:29:39 PM
TestCount=614
Status=Candidate certified prime

[Running Times]
Initialization=1mn 46s
1stPhase=2071h 22mn 47s
2ndPhase=563h 45mn 34s
Total=2635h 10mn 8s

[Candidate]
File=D:\Primo220\wrk\f1667\f1667.in
Expression=1667!-1667@2+2

Cheers
Ken
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