Return to Question

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 $-1$. 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.

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.

In McMuellen'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 $-1$. 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.

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 $-1$. 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.

In McMuellen'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 $-1$. 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.

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

In McMuellen'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 $-1$. 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.