Prime Number divides Infinite Number of Fibonacci Numbers
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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): $5$