Fibonacci Sequence


A Fibonacci sequence is easily constructed:  Start with 0 and 1, and for each following number, add the previous two: 0, 1, 0+1=1, 1+1=2, 1+2=3, and so on. Here is the beginning of the sequence:  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

The Fibonacci sequence is named after Leonardo of Pisa, who is also known as Fibonacci. In 1202 he wrote a book entitled “Liber Abaci“,  which introduces the sequence to Western European mathematics. It had been described earlier, however, in Indian mathematics. 

Fibonacci number sequences are everywhere in nature, and the math can quickly get very complex. Some numbers in the sequence are also prime, which creates an interesting way to look at primes, and more problems, for instance: Are there an infinite number of Fibonacci Primes?

Here is some math from the Wikipedia:

Fibonacci primes

“Main article: Fibonacci prime

Fibonacci prime is a Fibonacci number that is prime. The first few are:

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ….

Fibonacci primes with thousands of digits have been found, but it is not known whether there are infinitely many.[29]

Fkn is divisible by Fn, so, apart from F4 = 3, any Fibonacci prime must have a prime index. As there are arbitrarily long runs of composite numbers, there are therefore also arbitrarily long runs of composite Fibonacci numbers.

With the exceptions of 1, 8 and 144 (F1 = F2F6 and F12) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael’s theorem).[30]

144 is the only nontrivial square Fibonacci number.[31] Attila Pethő proved[32] in 2001 that there are only finitely many perfect power Fibonacci numbers. In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that only 8 and 144 are non-trivial perfect powers.[33]

No Fibonacci number greater than F6 = 8 is one greater or one less than a prime number.[34]

Any three consecutive Fibonacci numbers, taken two at a time, are relatively prime: that is,

gcd(FnFn+1) = gcd(FnFn+2) = 1.

More generally,

gcd(FnFm) = Fgcd(nm).

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