Etceterist : Beige Knight

Fibonacci sequence

Etceterist said Jun 28, 2006, 8:08 AM:

 

It's one of the simplest things.  Start with nothing, add one, then add them together, then keep adding the last two numbers.  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,…

Zero is the zeroth fibo, which makes 144 the 12th (twelfth?  twelveth?) and (I think) the only square in the sequence.  For sure, 144 is the highest fibo that does not have a novel prime number as a factor (the only other being the sixth fibo:  8).  Novel, here, means not being a factor of any fibo earlier in the sequence.   13 is itself prime; 21 has 7; 34 has 17; 55 has 11; 89 and 233 are both prime.  It goes on and on and on as infinitely long as the natural numbers themselves.

Some bizarrities rise up once you check deeper into it.  If the fibo's number in the sequence is prime, it's factors are all novel primes For examples, the 17th fibo is 1597, which is novel, the 53rd is 953 X 55945741, both of which are novel (the 106th is 953 X 55945741 X 119218851371, or the novel factors of the 53rd times its own novel prime).  263rd is 4733 X 93629 X some 46 digit monstrosity I'll spare you…these numbers get huge fast.   

There is something fundamentally comforting in this sequence for me.  It is a source of novelty, which is something I have wondered about for the Anyverse.  If there is no novelty, there is no free will and really where's the fun in that?  This simple algorithm returns constant and assured novelty for a countably infinite number of iterations. Anything is, therefore, possible.