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Let's consider the Fibonacci sequence, that is the sequence of naturals defined by:

$F_1=F_2=1$

$F_{n+1}=F_{n}+F_{n-1}$

It is an open problem whether the sequence contains an infinite number of primes.

It is known that, after $\,F_3=2\,$ and $\,F_4=3$, the only possible primes are those of the form:

$F_{6m-1}=F_{3m-1}^2+F_{3m}^2$

$F_{6m+1}=F_{3m}^2+F_{3m+1}^2$

One could expect that the number of primes of the form $\,F_{6m-1}\,$ was similar to the number of primes of the form $\,F_{6m+1}$, but brute force shows that:

up to $\,m=10$, there are $\,6\,$ primes of the form $\,F_{6m-1}\,$ and only $\,3\,$ of the form $\,F_{6m+1}$;

up to $\,m=100$, there are $\,14\,$ primes of the form $\,F_{6m-1}\,$ and only $\,5\,$ of the form $\,F_{6m+1}$;

up to $\,m=1000$, there are $\,15\,$ primes of the form $\,F_{6m-1}\,$ and only $\,7\,$ of the form $\,F_{6m+1}$;

up to $\,m=5522$, there are $\,17\,$ primes of the form $\,F_{6m-1}\,$ and only $\,10\,$ of the form $\,F_{6m+1}$.

How can this asymmetric behavior be explained?

Thanks.

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    $\begingroup$ $m=1000$ is a very small number in asymptotics, even given the size of $F_m$. Could there be some sort of Chebyshev bias going on? $\endgroup$ Mar 23, 2020 at 22:34
  • $\begingroup$ I don't know ... the computation is much time-expensive ... my workstation is working on $\,m=10000\,$ ... $\endgroup$ Mar 23, 2020 at 22:36
  • $\begingroup$ It seems appropriate to gather a numerically significant amount of data before asking for explanation of possible coincidences. (For that matter, one could also ask why the asymmetric behaviour needs an explanation—the only justification given for the symmetry was "One could expect" it.) $\endgroup$
    – LSpice
    Mar 23, 2020 at 22:41
  • $\begingroup$ Well, in the context of natural numbers the same simmetry of distribution would be a consequence of Dirichlet's theorem on arithmetic progressions ... $\endgroup$ Mar 23, 2020 at 22:46
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    $\begingroup$ There is already a lot of fine bias (i.e. asymmetry) in the distribution of all prime numbers in residue classes. I suggest that you study the relevant literature, e.g. en.wikipedia.org/wiki/Chebyshev%27s_bias and arxiv.org/abs/1603.03720 $\endgroup$
    – GH from MO
    Mar 23, 2020 at 23:45

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