Where Fibonacci numbers surpass prime numbers

[Loren Booda]

The series of prime numbers p_n=2, 3, 5, 7, 11, 13, 17, 19, 23, 27..., and Fibonacci numbers F_n=0, 1, 1, 2, 3, 5, 8, 13, 21, 34..., suggest that F_n might be considered to surpass p_n exactly at an irrational value n_s such that 9<n_s<10 and can be determined most exactly from both series as n-->infinity.

How would you determine n_s?

 

[CRGreathouse]

There's a nice closed-form for the Fibonacci numbers, but there's nothing so nice for the primes that extends them continuously and 'nicely' to the noninteger reals. So I wouldn't know of a good way to do this.

 

[Loren Booda]

Thanks for your contribution, CRGreathouse. You seem to have addressed the heart of my problem.

 

[Office_Shredder]

It seems like you want to describe the primes and the fibonnaci numbers as some functions p(n) and f(n) and then extend those functions to the real numbers...how would you do this?

 

[rscosa]

Hi!
I think your conjecture is wrong although to find a counterexample you need to go so so far away. The Fibonacci sequence satisfies the recurrence relation: $F_n=F_{n-1}+F_{n-2}$ with $F_1=1$ (or $0$ depends how you define it but it does not matter). Now, if you consider the recurrence relation: $L_n=L_{n-2}+L_{n-3}$ (looks like similar) with initial conditions $L_1=0$, $L_2=2$, $L_3=3$ it is 'simple' (you need some mathematic's background) to proof that if N is prime $L_N$ is also prime but the reverse is not true but to find a counterexample you need to go, as I said before, so far away, indeed it is possible to find a prime P such that $L_{P^2}$ is prime but this number is large but, of courseit , is possible to compute. By the way this last sequence I think is called Lucas sequence.