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November 12th, 2007, 06:22 PM   #1
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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)?
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November 12th, 2007, 06:46 PM   #2
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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 counter-example is that 4181 is the 19th term in the sequence if you start with 1, but 4181 = 37*113 and thus is not prime.
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November 12th, 2007, 06:51 PM   #3
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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.
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November 12th, 2007, 07:05 PM   #4
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Ah, thanks for the correction. In that case, take a look at this link: http://mathworld.wolfram.com/FibonacciPrime.html
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November 13th, 2007, 03:17 PM   #5
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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).
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November 13th, 2007, 03:54 PM   #6
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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.
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November 13th, 2007, 05:21 PM   #7
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Quote:
Originally Posted by soandos
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).
A_n = 6n - 3 is an additive function that produces an infinite number of odd numbers, but only one prime. It's actually not obvious when an infinite sequence contains an infinite number of primes -- look at Mersenne numbers for an example. Lots of people think that there are infinitely many Mersenne primes, even though we only know of 40-some. But there's been no proof thus far.
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November 13th, 2007, 07:10 PM   #8
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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 no-one 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?
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November 14th, 2007, 06:22 AM   #9
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F(a) and F(b) are divisors of F(ab).
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November 14th, 2007, 06:37 AM   #10
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Quote:
Originally Posted by soandos
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 no-one is sure whether it is prime or not, but it has passed some primality tests.
I did some quick calculations, and it looks to me like the Mersenne.org project has put more than 21 million PII-400 years into its search for Mersenne primes. If that's a lack of processing power, what wouldn't be?

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:
Originally Posted by soandos
all this bieng said, what would the proof involve to show that there are an infinite amount of primes (merrisane or Fibonacci).
Who knows? Until someone publishes a proof like that I wouldn't have a clue. It's not at all obvious that it *could* be proved.

Quote:
Originally Posted by soandos
and what was the proof that all Fibonacci primes have a prime index?
The proof that Mersenne primes must have a prime exponent:
Suppose 2^(ab)-1 is prime, where a,b > 1. Then 2^a-1 and 2^b-1 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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