From moree@mpim-bonn.mpg.de Wed Oct 15 20:39:47 1997
Date: Wed, 15 Oct 1997 13:46:25 -0400
From: Pieter Moree <moree@mpim-bonn.mpg.de>
To: NMBRTHRY@LISTSERV.NODAK.EDU
Subject: Artin conjecture for Fibonacci numbers

Fred Lunnon wrote the following concerning the
`rank of apparition' of primes in Fibonacci numbers:

>From a remark of Nelson Stephens, pace Artin's conjecture, which I guess must
>have something to do with primitive roots being uniformly distributed, it's
>a reasonable guess that the probability of F(p +/- 1) being the first one
>divisible by prime p is just 6/pi^2 = 0.607927, in good agreement with the
>61% observed for both cases of p mod 5.

This is not a reasonable guess.
Artin's conjecture states that every integer not equal
to -1 or a square is a primitive root modulo p for
infinitely many p and proposes a density for the set of such p.
These densities are always rational multiples op Artin's constant
A=\prod (1-1/(q(q-1)), where q ranges over the primes.
Numerically, A=.37395 58136 ....

Under the assumption of the Generalized Riemann Hypothesis Artin's conjecture
was solved in 1967 by C. Hooley.

Artin's argument also works in the Fibonacci case.
Writing r=(1+sqrt 5)/2 for the golden ratio, the prime p
first appears in the [p - (5/p)]-th Fibonacci number
when -r^2 has order p - (5/p) modulo p.

For the primes p=1,4 mod 5, which have (5/p)=1, it means
that (-r^2 mod p) is a primitive root in F_p^*.
This does not depend on the choice of (-r^2 mod p) in F_p^*.
Analysis as for Artin's conjecture shows that for our p=1,4 mod 5
the Euler factor 1-1/(q(q-1)=19/20 at q=5 needs to be replaced by
1-2/(q(q-1)=18/20. This implies that for a fraction 18A/19 of all
primes p=1,4 mod 5 we find that (-r^2 mod p) has order p-1.


For the primes p=2,3 mod 5, which have (5/p)=-1, we want
that (-r^2 mod p) has order p+1 in the multiplicative group
of the field F_p(r) of order p^2. [From norm(-r^2)=1 we see
that the order always divides p+1.]
In this case the analysis is a bit more subtle as we have to combine
Artin's splitting conditions with an inertia condition. The conclusion
is that we have to leave out the Euler factor 19/20 at q=5.
So for a fraction 20A/19 of all primes p=2,3 mod 5 we should find
that (-r^2 mod p) has order p+1.

To prove these results, we need to assume, as did Hooley, the
Generalized Riemann Hypothesis.

Pieter Moree
Peter Stevenhagen.

PS. Using PARI we approximated the
proposed densities numerically, with the kind help of

Karim Belabas,

and our findings are consistent with the densities we claim.
The convergence seems to be rather slow, unfortunately.

18A/19 is approximately 0.354273928691875851841321314
20A/19 is approximately 0.393637698546528724268134794