For the distribution of prime numbers there is a hypothesis which predicts the possible positions of prime numbers called Riemann hypothesis
Using Fibonacci it is also possible to 'predict' prime number positions by calculating the
gcd of a Fibonacci number and a prime number.
From a computational standpoint it makes no sense calculating the
gcd of a larger number solely to know the
gcd of a smaller number.
But is it true that from a mathematical point of view the primality property of the "position" is related to the Fibonacci number itself? I understand that "position" in a set is a discrete value. But Fibonacci seems to make these values more continuous as they increase. Or is it in no way possible to relate an index number to an actual value?
if x is prime then x = GCD(x, fibonacci(x-1)) or GCD(x, fibonacci(x+1))
What interests me most is the
true which the above method returns for numbers which aren't prime numbers or prime numbers which aren't generated (I guess these are pseudoprimes, but I'm not sure on the exact definition).
Has the primal property of Fibonacci numbers and the seemingly 'lesser and lesser occurrence of pseudoprimes as values increase' been related to that same kind of 'increasingly better predictive behaviour as the values increase' seen in the Riemann hypothesis?
Edit: Or is this what Gauss and Riemann discussed?
Edit 2: More specific question for finding solution
The Riemann hypothesis: https://www.youtube.com/watch?v=yhtcJPI6AtY#t=2485
Fourier transforms of Von Mangoldt function to produce spikes: http://en.wikipedia.org/wiki/Von_Mangoldt_function
Now the Fibonacci pseudoprimes distribution also generates a certain 'pattern' (i've visualized it in the picture below). The pattern behaves the same as the Von Mangoldt function in the sense that the values stay within the horizontal hyperbole.
Thanks to Gerry Myerson I'm able to explain what I'm doing:
let's call the Fibonacci pseudoprimes $ai$, so $a0=1$, $a1=323$, $a2=377$, and so on. for each $n$ ,$n=1,2,…,$ I am plotting the point $pn=(n,an−an−1)$, and then I am drawing the line segment joining $pn$ to $pn+1$ for $n=1,2,…$
"Fibonacci pseudoprimes produced 'random' pattern:"
Edit: Better graph for
Old graph (shows
Is it possible and does it make sense to apply the Fourier transforms mentioned above to the Fibonacci-pseudoprime pattern instead of the Von Mangoldt function?