Subfield: Number Theory


Question: Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $\operatorname{gcd}\left(a_{i}, a_{i+1}\right)>a_{i-1}$. Prove that $a_{n} \geq 2^{n}$ for all $n \geq 0$.

Solution:
Since $a_{i} \geq \operatorname{gcd}\left(a_{i}, a_{i+1}\right)>a_{i-1}$, the sequence is strictly increasing. In particular $a_{0} \geq 1, a_{1} \geq 2$. For each $i \geq 1$ we also have $a_{i+1}-a_{i} \geq \operatorname{gcd}\left(a_{i}, a_{i+1}\right)>a_{i-1}$, and consequently $a_{i+1} \geq a_{i}+a_{i-1}+1$. Hence $a_{2} \geq 4$ and $a_{3} \geq 7$. The equality $a_{3}=7$ would force equalities in the previous estimates, leading to $\operatorname{gcd}\left(a_{2}, a_{3}\right)=\operatorname{gcd}(4,7)>a_{1}=2$, which is false. Thus $a_{3} \geq 8$; the result is valid for $n=0,1,2,3$. These are the base cases for a proof by induction.

Take an $n \geq 3$ and assume that $a_{i} \geq 2^{i}$ for $i=0,1, \ldots, n$. We must show that $a_{n+1} \geq 2^{n+1}$. Let $\operatorname{gcd}\left(a_{n}, a_{n+1}\right)=d$. We know that $d>a_{n-1}$. The induction claim is reached immediately in the following cases:

$$
\begin{array}{ll}
\text { if } \quad a_{n+1} \geq 4 d & \text { then } a_{n+1}>4 a_{n-1} \geq 4 \cdot 2^{n-1}=2^{n+1} \\
\text { if } \quad a_{n} \geq 3 d & \text { then } a_{n+1} \geq a_{n}+d \geq 4 d>4 a_{n-1} \geq 4 \cdot 2^{n-1}=2^{n+1} \text {; } \\
\text { if } a_{n}=d \quad \text { then } a_{n+1} \geq a_{n}+d=2 a_{n} \geq 2 \cdot 2^{n}=2^{n+1}
\end{array}
$$

The only remaining possibility is that $a_{n}=2 d$ and $a_{n+1}=3 d$, which we assume for the sequel. So $a_{n+1}=\frac{3}{2} a_{n}$.

Let now $\operatorname{gcd}\left(a_{n-1}, a_{n}\right)=d^{\prime}$; then $d^{\prime}>a_{n-2}$. Write $a_{n}=m d^{\prime} \quad(m$ an integer $)$. Keeping in mind that $d^{\prime} \leq a_{n-1}<d$ and $a_{n}=2 d$, we get that $m \geq 3$. Also $a_{n-1}<d=\frac{1}{2} m d^{\prime}$, $a_{n+1}=\frac{3}{2} m d^{\prime}$. Again we single out the cases which imply the induction claim immediately:

if $m \geq 6$ then $a_{n+1}=\frac{3}{2} m d^{\prime} \geq 9 d^{\prime}>9 a_{n-2} \geq 9 \cdot 2^{n-2}>2^{n+1} ;$

if $3 \leq m \leq 4$ then $a_{n-1}<\frac{1}{2} \cdot 4 d^{\prime}$, and hence $a_{n-1}=d^{\prime}$,

$$
a_{n+1}=\frac{3}{2} m a_{n-1} \geq \frac{3}{2} \cdot 3 a_{n-1} \geq \frac{9}{2} \cdot 2^{n-1}>2^{n+1} \text {. }
$$

So we are left with the case $m=5$, which means that $a_{n}=5 d^{\prime}, a_{n+1}=\frac{15}{2} d^{\prime}, a_{n-1}<d=\frac{5}{2} d^{\prime}$. The last relation implies that $a_{n-1}$ is either $d^{\prime}$ or $2 d^{\prime}$. Anyway, $a_{n-1} \mid 2 d^{\prime}$.

The same pattern repeats once more. We denote $\operatorname{gcd}\left(a_{n-2}, a_{n-1}\right)=d^{\prime \prime}$; then $d^{\prime \prime}>a_{n-3}$. Because $d^{\prime \prime}$ is a divisor of $a_{n-1}$, hence also of $2 d^{\prime}$, we may write $2 d^{\prime}=m^{\prime} d^{\prime \prime}$ ( $m^{\prime}$ an integer). Since $d^{\prime \prime} \leq a_{n-2}<d^{\prime}$, we get $m^{\prime} \geq 3$. Also, $a_{n-2}<d^{\prime}=\frac{1}{2} m^{\prime} d^{\prime \prime}, \quad a_{n+1}=\frac{15}{2} d^{\prime}=\frac{15}{4} m^{\prime} d^{\prime \prime}$. As before, we consider the cases:

$$
\text { if } m^{\prime} \geq 5 \quad \text { then } a_{n+1}=\frac{15}{4} m^{\prime} d^{\prime \prime} \geq \frac{75}{4} d^{\prime \prime}>\frac{75}{4} a_{n-3} \geq \frac{75}{4} \cdot 2^{n-3}>2^{n+1}
$$

$$
\text { if } 3 \leq m^{\prime} \leq 4 \text { then } a_{n-2}<\frac{1}{2} \cdot 4 d^{\prime \prime} \text {, and hence } a_{n-2}=d^{\prime \prime} \text {, }
$$

$$
a_{n+1}=\frac{15}{4} m^{\prime} a_{n-2} \geq \frac{15}{4} \cdot 3 a_{n-2} \geq \frac{45}{4} \cdot 2^{n-2}>2^{n+1}
$$

Both of them have produced the induction claim. But now there are no cases left. Induction is complete; the inequality $a_{n} \geq 2^{n}$ holds for all $n$.