In McMuellen's [Uniformly Diophantine numbers in a fixed real quadratic field][1] 
generalized Fibonacci sequence are defined as follows:

$f_0=0,f_1=1,f_m=tf_{m-1}-nf_{m-2}$ where some fixed $t\in \mathbb Z$ and $n$ is $+1$ or $-1$.
For example, for $t=1,n=-1$ we get the usual Fibonacci sequence.

My question: Does there exist $t,n$ such that the resulting Fibonacci sequence has infinitely primes in it?
I think that it is conjectured to hold for the usual Fibonacci sequence.
A weaker assertion: Does there exist $t,n$ such that the resulting Fibonacci has infinitely many elements with a large prime divisor, e.g., infinitely many $m$'s such that $p_m|f_m$, $p_m$ prime and $\frac{p_m}{f_m}>C$ for some C>0?

An related paper (which does not contain any answer to the above questions) is 
(By Y. Bugeaud F. Luca, M. Mignotte et S. Siksek) [On Fibonacci numbers with few prime divisors][2]. 

I'll be happy to know about any reference that deal with these generalized sequences.



  [1]: http://www.math.harvard.edu/~ctm/papers/home/text/papers/cf/cf.pdf
  [2]: http://www-irma.u-strasbg.fr/~bugeaud/travaux/OmegaDef.pdf