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Old 2017-10-25, 07:42   #1
Shen
 
Jan 2017

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Smile GIMPS and Fibonacci prime project?

I have run prime95 for several years, as many of you, I didn't find one prime, that makes me a little disappointed.

At the same time, I do some research on Fibonacci prime, it seems a little easier to find one prime or PRP. There are 15 PRP and 34 primes , 49 totally until now.

Could someone set up a project for Fibonacci prime like GIMPS?
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Old 2017-10-25, 10:46   #2
science_man_88
 
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Quote:
Originally Posted by Shen View Post
I have run prime95 for several years, as many of you, I didn't find one prime, that makes me a little disappointed.

At the same time, I do some research on Fibonacci prime, it seems a little easier to find one prime or PRP. There are 15 PRP and 34 primes , 49 totally until now.

Could someone set up a project for Fibonacci prime like GIMPS?
you already can find they must be in specific positions, when done mod 6 the fibonacci sequence is 0,1,1,2,3,5,2,1,3,4,1,5,0,5,5,4,3,1,4,5,3,2,5,1,0,1,1,2,... repeats. So, the indexes that could be primes greater than 3 are 1,2,5,7,10,11,13,14,17,19,22,23 mod 24.
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Old 2017-10-25, 13:50   #3
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Quote:
Originally Posted by science_man_88 View Post
you already can find they must be in specific positions, when done mod 6 the fibonacci sequence is 0,1,1,2,3,5,2,1,3,4,1,5,0,5,5,4,3,1,4,5,3,2,5,1,0,1,1,2,... repeats. So, the indexes that could be primes greater than 3 are 1,2,5,7,10,11,13,14,17,19,22,23 mod 24.
Since

Fa | Fab,

Fa > 1 for a > 2,

F2k = Fk*Lk, and Lk > 1 for k > 1,

it is clear that Fn can be prime only if n = 4, or if n is prime.

I did find a page of Possible Fibonacci Primes listing some large Fibonacci PRP's, but it hasn't been updated in 11 years...
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Old 2017-10-25, 14:13   #4
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Originally Posted by Dr Sardonicus View Post
I did find a page of Possible Fibonacci Primes listing some large Fibonacci PRP's, but it hasn't been updated in 11 years...
This extends that list: http://www.primenumbers.net/prptop/s...&action=Search
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Old 2017-10-25, 15:35   #5
fivemack
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The problem is that Fibonacci numbers aren't the right shape for group-theoretic primality proofs, and nobody's seriously interested in finding large PRPs.
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Old 2017-10-25, 15:39   #6
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Originally Posted by Dr Sardonicus View Post
I did find a page of Possible Fibonacci Primes listing some large Fibonacci PRP's, but it hasn't been updated in 11 years...
Fibonacci (104911) is prime, see http://factordb.com/index.php?query=I%28104911%29 . The other 3 PRPs listed are still PRP in factordb.

Chris
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Old 2017-10-25, 20:59   #7
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Quote:
Originally Posted by fivemack View Post
The problem is that Fibonacci numbers aren't the right shape for group-theoretic primality proofs, and nobody's seriously interested in finding large PRPs.
People are, however, actively working on composite factors of Fibonacci and Lucas numbers that are, perhaps barely, within the range of present factoring methods.

See, e,g, posts 2201 and 2205 to the thread NFS@Home.
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