Lamé's Theorem/Lemma

Theorem
Let $a, b \in \Z_{>0}$ be (strictly) positive integers.

Let the Euclidean Algorithm be employed to find the GCD of $a$ and $b$. Suppose it takes $n$ cycles around the Euclidean Algorithm to find $\gcd \set {a, b}$.

Then $\min \set {a, b} \ge F_{n + 2}$, where $F_n$ denotes the $n$-th Fibonacci number.

Proof
suppose $a \ge b$.

Let $q_i, r_i$ be the quotients and remainders of each step of the Euclidean Algorithm, that is:

so $r_n = \gcd \set {a, b}$.

We prove that $r_{n - m} \ge F_{m + 1}$ for $0 \le m < n$ by induction:

Basis for the Induction
When $m = 0$, we have:

When $m = 1$, we have:

These are our base cases.

Induction Hypothesis
This is our induction hypothesis:
 * $r_{n - k + 2} \ge F_k$
 * $r_{n - k + 1} \ge F_{k + 1}$

for some $k$ where $2 \le k < n$.

It is to be demonstrated that:
 * $r_{n - k} \ge F_{k + 2}$

Induction Step
This is our induction step:

We have shown that the result for $m = k - 2$ and $m = k - 1$ implies the result for $m = k$.

Thus we have $r_{n - m} \ge F_{m + 2}$ for all $0 \le m < n$.

Hence: