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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.

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  • $\begingroup$ 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. $\endgroup$
    – Charles
    Commented Nov 26, 2011 at 18:58
  • $\begingroup$ @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. $\endgroup$
    – Menny
    Commented Nov 27, 2011 at 9:27

2 Answers 2

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These are Lucas Sequences (http://en.wikipedia.org/wiki/Lucas_sequence), of which the Fibonacci Sequence is a specific case, and they share with the usual Fibonacci Sequence the following characteristics which, unless there are further reasons preventing what I'm about to say from being true, would allow the same sort of heuristic analysis for primes in these sequences as that appearing for Fibonacci Primes in Section 5 of the paper by Bugeaud, Luca, Mignotte and Siksek you mentioned.

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.

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  • $\begingroup$ Thanks Timothy - I always thought Lucas sequence is what Wikipedia calls "Lucas numbers". $\endgroup$
    – Menny
    Commented Nov 28, 2011 at 13:53
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Try $t=2$, $n=1$ where $f_m = m$.

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  • $\begingroup$ @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). $\endgroup$
    – Menny
    Commented Nov 27, 2011 at 9:37

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