# Primes in generalized fibonacci sequences

In C. McMullen's Uniformly Diophantine numbers in a fixed real quadratic field 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 $-1$ and $t^2-4n>0$. 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?

A 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.

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

• I'm not sure that the weaker assertion is much easier, since it comes down to asking if there are infinitely many terms of the form $f_m=p_ms$ with s in some fixed finite set (rather than just {1}). If you had asked for $\log(p_m)/\log(f_m)>0.99$ that would be easier. Nov 26, 2011 at 18:58
• @Charles: note that I wrote e.g. after "large prime divisor" - your concept of large is also very interesting; I'll be very happy to hear any thought on how to approach it. Nov 27, 2011 at 9:27

a)Binet-type formula, which, apart from special cases like $t=2, n=1$ mentioned by Robert Israel, give exponential growth.
b)According to Wikipedia, the $m$-th term divides the $mk$-th term, so only those with prime indices have a chance to be prime.
Try $t=2$, $n=1$ where $f_m = m$.
• @Robert: thanks for your answer. As in Mcmullen's article my motivation is about sequences with $t^2-4n>0$ (which is now changed above). Nov 27, 2011 at 9:37