Prime Number divides Fibonacci Number

Theorem

For $n \in \Z$, let $F_n$ denote the $n$th Fibonacci number.

Let $p$ be a prime number.

Then:

$p \equiv \pm 1 \pmod 5 \implies p \divides F_{p - 1}$

$p \equiv \pm 2 \pmod 5 \implies p \divides F_{p + 1}$

where $\divides$ denotes divisibility.

Thus in all cases, except where $p = 5$ itself:

$p \divides F_{p \pm 1}$

Proof

It is worth noting the one case where $p = 5$:

$5 \divides F_5 = 5$

This theorem requires a proof. In particular: Googling around suggests there is a proof based on the Law of Quadratic Reciprocity but I have not laid hands on it yet. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$