GCD of Fibonacci Numbers

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Theorem

Let $F_k$ be the $k$th Fibonacci number.


Then:

$\forall m, n \in \Z_{> 2}: \gcd \set {F_m, F_n} = F_{\gcd \set {m, n} }$

where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.


Proof

From the initial definition of Fibonacci numbers, we have:

$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$

Without loss of generality, let $m \le n$.

Let $h$ be $\gcd \set {m, n}$.

Let $a$ and $b$ be integers such that $m = h a$ and $n = \map h {a + b}$.

$a$ and $a + b$ are coprime by Integers Divided by GCD are Coprime.

Therefore, $a$ and $b$ are coprime by Integer Combination of Coprime Integers.

\(\ds \gcd \set {F_m, F_n}\) \(=\) \(\ds \gcd \set {F_{h a}, F_{h a - 1} F_{h b} + F_{h a} F_{h b + 1} }\) Honsberger's Identity
\(\ds \) \(=\) \(\ds \gcd \set {F_{h a}, F_{h a - 1} F_{h b} }\) GCD with Remainder

Let $u$ and $v$ be integers such that $F_{h a} = u F_h$ and $F_{h b} = v F_h$, whose existence is proved by Divisibility of Fibonacci Number.

We have that $F_{h a}$ and $F_{h a - 1}$ are coprime by Consecutive Fibonacci Numbers are Coprime.

Therefore, $u$ and $F_{h a - 1}$ are coprime by Divisor of One of Coprime Numbers is Coprime to Other.

\(\ds \gcd \set {F_{h a}, F_{h a - 1} F_{h b} }\) \(=\) \(\ds F_h \gcd \set {u, v F_{h a - 1} }\) Honsberger's Identity
\(\ds \) \(=\) \(\ds F_h \gcd \set {u, v}\) Solution of Linear Diophantine Equation
\(\ds \) \(=\) \(\ds \gcd \set {F_m, F_{n - m} }\)

Therefore:

$\forall m, n \in \Z_{>2} : \gcd \set {F_m, F_n} = \gcd \set {F_m, F_{n - m} }$

This can be done recurrently to produce the result, in a fashion similar to the Euclidean Algorithm.

Since $a$ and $b$ are coprime, the result would be $\gcd \set {F_h, F_h}$.


Therefore:

$\forall m, n > 2 : \gcd \set {F_m, F_n} = F_{\gcd \set {m, n} }$

$\blacksquare$


Historical Note

This result was reported by François Édouard Anatole Lucas in $1876$, according to Donald E. Knuth in his The Art of Computer Programming: Volume 1: Fundamental Algorithms, 3rd ed..

The specific outlet in which he published this is being sought.


Sources

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