Prime Number divides Infinite Number of Fibonacci Numbers

Theorem

Let $p$ be a prime number.

Then there exist an infinite number of Fibonacci numbers which are divisible by $p$.

Proof

From Prime Number divides Fibonacci Number, either $F_{p - 1}$ or $F_{p + 1}$ is divisible by $p$.

Thus:

\(\ds p\)

\(\divides\)

\(\ds F_{p \pm 1}\)

Prime Number divides Fibonacci Number

\(\ds \forall n \in \Z_{>0}: \, \)

\(\ds F_{p \pm 1}\)

\(\divides\)

\(\ds F_{n \paren {p \pm 1} }\)

Divisibility of Fibonacci Number

\(\ds \leadsto \ \ \)

\(\ds \forall n \in \Z_{>0}: \, \)

\(\ds p\)

\(\divides\)

\(\ds F_{n \paren {p \pm 1} }\)

$\blacksquare$

Sources

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