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Old 2015-07-13, 23:30   #5
Batalov
 
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"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

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Unfortunately, no derivative primes were found.

Here are the examples of the tested candidates to give you a flavor of what was done:
Code:
Phi(10,(46421883*2^198800-1)*(299771367*2^198800-1)) is composite: RES64: [DB6C0836F42F050D]
Phi(5,(125526867*2^198800-1)*(299771367*2^198800-1)-1) is composite: RES64: [81248E7866F4AD15]
and a few thousand more...
There are a 200K of similar canidates easily produced and some of them are likely prime and can be found 
but the cost of sieving and testing is too high to go on for too long. 
Besides I already have a couple of primes like these.
What was of primary interest, though, and what I 100% searched were all Phi(5,p), Phi(7,p), Phi(10,p+1), Phi(14,p+1) for all of your several hundred p.
These are more interesting and rare.
They are provable by N-1 because p and p+1 are fully factored which makes
Phi(5,...)-1 and Phi(10,...)-1 50% factored and Phi(7,...)-1 and Phi(14,...)-1 33.333% factored which is enough got the proof.
Other polynomials can be tried (and tried and tried until a constructible prime is found) but they are not as interesting as Phi().
I have one of these but it was not eligible for Top5000. Just a little too small.
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