asher63 (asher63) wrote,

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Fibonacci Primes

You know that prime numbers are cool. And you know that Fibonacci numbers are way cool. But did you know just how cool Fibonacci primes are?

Fibonacci numbers, you know, are numbers in the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ... where each successive number is the sum of the two before it. And primes can't be divided by any number except themselves. Fibonacci primes, as I'm sure you've already guessed, are Fibonacci numbers that are also prime.

But wait! Here's the super cool part. With the exception of the number 3, every known Fibonacci prime occupies a prime-numbered place in the sequence of Fibonacci numbers!!! No, really, I'm not making this up. Look, the first few Fibonacci primes are {2, 3, 5, 13, 89, 233, 1597 ...}. These are numbered third, fourth (there's that solitary exception), fifth, seventh, eleventh, and 13th on the Fibonacci sequence. The next number, 1597, comes in at 17th. And the pattern holds for as far as mathematicians are able to test it. Here's the Wolfram Research article on Fibonacci Primes:
The first few proven prime Fibonacci numbers are F(n) are 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... (Sloane's A005478), which occur for n=3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839, 104911, 130021, 148091, 201107, 397379, ... (Sloane's A001605; Dubner and Keller 1999), where the Fibonacci numbers with indices 104911 (B. de Water), 130021 (D. Fox), 148091 (T. D. Noe) and 201107, 397379, 433781, 590041, 593689, and 604711 (H. Lifchitz) are probable primes (Caldwell).

The article reminds us that "the converse is not true" and some numbers holding prime spots on the Fibonacci chain aren't prime themselves (the 19th Fibonacci number, 4181, isn't prime.)

But still. Cool, or what?

Well, maybe only for number geeks.
Tags: geekery

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