Let $\{F_n\}, n\in \mathbb{N}$ be the sequence of Fibonacci numbers such that $F_1=1$, $F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ $\forall n\geq2$.

Define a new sequence $\{S_n\}$ such that $S_n=F_n+k$ $\forall n\in \mathbb{N}$ and $k$ is a fixed natural number.

Now the question is: What are the values of $k$ such that, for every prime $p$, there exist an $N\in \mathbb{N}$, such that $p|S_N$ ?

Can we give a pattern for all such values?

  • $\begingroup$ Can we have at least one such value beyond 0, to begin with? $\endgroup$ – Ivan Neretin May 26 '17 at 11:28
  • $\begingroup$ Yes k=2 is one of them $\endgroup$ – Arpan1729 May 26 '17 at 14:00

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