November 12th, 2007, 06:22 PM  #1 
Member Joined: Oct 2007 Posts: 68 Thanks: 0  fibonacci and prime number correlation
does every prime Fibonacci occur on a prime term, other than 3 (if one starts with 1,1,2, not 0,1,1,2)?

November 12th, 2007, 06:46 PM  #2 
Senior Member Joined: Dec 2006 Posts: 1,111 Thanks: 0 
No, that's actually not true, although it does hold true for a good while in the first few terms of the sequence. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181... A counterexample is that 4181 is the 19th term in the sequence if you start with 1, but 4181 = 37*113 and thus is not prime. 
November 12th, 2007, 06:51 PM  #3 
Senior Member Joined: Nov 2007 Posts: 258 Thanks: 0 
Infinity, I believe you misread the question. The OP asked if every prime number F(n) has a prime index, not if every number F(n) is prime when the index is.

November 12th, 2007, 07:05 PM  #4 
Senior Member Joined: Dec 2006 Posts: 1,111 Thanks: 0 
Ah, thanks for the correction. In that case, take a look at this link: http://mathworld.wolfram.com/FibonacciPrime.html 
November 13th, 2007, 03:17 PM  #5 
Member Joined: Oct 2007 Posts: 68 Thanks: 0  thanks
thank you. just a question on the page though. how could there not be an infinite amount of Fibonacci primes? after all, it is an additive function that over its course will produce an infinite amount of odd numbers. why would that not mean that since it has already been shown to have some primes, would it not continue, as the set of all primes has already been proven infinite (so there is no upper bound for the Fibonacci primes, allowing an infinite amount). 
November 13th, 2007, 03:54 PM  #6 
Senior Member Joined: Dec 2006 Posts: 1,111 Thanks: 0 
Well, it would seem rather intuitively obvious that there are an infinite number of Fibbonaci primes, and indeed it is assumed to be probably true; However, keep in mind that the number of primes per a given amount of Fibbonaci numbers keeps decreasing as you keep getting further and further along, so it's also possible that they eventually vanish completely, leaving only composite numbers left.

November 13th, 2007, 05:21 PM  #7  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: thanks Quote:
 
November 13th, 2007, 07:10 PM  #8 
Member Joined: Oct 2007 Posts: 68 Thanks: 0  replies
to infinity, even if the amount of Fibonacci primes does decrease asymptotically, does that mean that there are still an infinite amount? since the amount is clearly not decreasing linearly, and they are getting rarer it would seem that it is asymptotic. i maybe wrong (very probable actually, i am in 10th grade, with just miscellanea to go on for things like this), but it seems likely that given the fact that there are not a known number of finite Fibonacci primes, it seems, given the above, that there are infinite. to CRGreathouse, i meant a function that did not have a clearly finite amount of primes in it. and while there may only be ~40 merrisane primes, i believe that that is due to a lack of processing power. for example there are many primes that are considered "probable" as noone is sure whether it is prime or not, but it has passed some primality tests. all this bieng said, what would the proof involve to show that there are an infinite amount of primes (merrisane or Fibonacci). and what was the proof that all Fibonacci primes have a prime index? 
November 14th, 2007, 06:22 AM  #9 
Global Moderator Joined: Dec 2006 Posts: 18,702 Thanks: 1527 
F(a) and F(b) are divisors of F(ab).

November 14th, 2007, 06:37 AM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: replies Quote:
I'm not aware of any indices for Mersenne numbers that have passed any kind of primality test but are not known to be prime. Mersenne numbers, being of a very special form, are far easier to test than "random" numbers of a similar size. Quote:
Quote:
Suppose 2^(ab)1 is prime, where a,b > 1. Then 2^a1 and 2^b1 divide 2^(ab)1, so 2^(ab)1 is composite. For Fibonacci primes, the article http://en.wikipedia.org/wiki/Fibonacci_prime mentions a rule for the GCD of Fibonacci numbers which rules out prime Fibonacci numbers at nonprime indices > 4.  

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