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.
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Sources

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