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.