Are there infinitely many prime Fibonacci numbers?
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Probably yes. Not proved yet. However, it was proved that the only Fibonacci numbers that are also members of the set of twin primes are 3, 5, and 13. A twin prime is a prime number that is either 2 less or 2 more than another prime number. E.g. (11, 13)
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More interestingly: 1) Every 3rd Fibonacci number is even 2) Fn divides Fmn ∴ ∀ Fibonacci prime must have prime index (converse isn't true) Exception F4=3 Largest proven Fibonacci prime = F104911 (104911 is prime) Largest probable Fibonacci prime = F3244369 (3244369 is prime)
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How do you prove a number is prime
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If a number has no prime divisors smaller than its root, it kust be prime. For example, the square root of 17 is a little over 4, so if you show 17/2 isn't a whole number, and that 17/3 isn't a whole number, 17 must be prime
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Do you know the Fibonacci sequence 1,1,2, 3,5,8, 13, 21 ... ? Do you know what a prime number is (2,3,5,7,11,13,17,19)? OP wrote F_n to mean the nth Fibonacci number. For example, F_6 = 8. The claim then is that F_104911 is prime and this is the largest Fib prime we know.
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Interestingly this is also the number to call in Siena for prime, rich pasta Alfredo and spaghetti al forno. Fibonacci Kitchen is where you get a meal like Mama used to make. If your Mama was a mathematician.
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