[Number Theory]
Problem: Let $m>1$ be an integer. A sequence $a_{1}, a_{2}, a_{3}, \ldots$ is defined by $a_{1}=a_{2}=1$, $a_{3}=4$, and for all $n \geq 4$,

$$
a_{n}=m\left(a_{n-1}+a_{n-2}\right)-a_{n-3} .
$$

Determine all integers $m$ such that every term of the sequence is a square.

Solution: Consider an integer $m>1$ for which the sequence defined in the problem statement contains only perfect squares. We shall first show that $m-1$ is a power of 3 .

Suppose that $m-1$ is even. Then $a_{4}=5 m-1$ should be divisible by 4 and hence $m \equiv 1(\bmod 4)$. But then $a_{5}=5 m^{2}+3 m-1 \equiv 3(\bmod 4)$ cannot be a square, a contradiction. Therefore $m-1$ is odd.

Suppose that an odd prime $p \neq 3$ divides $m-1$. Note that $a_{n}-a_{n-1} \equiv a_{n-2}-a_{n-3}(\bmod p)$. It follows that modulo $p$ the sequence takes the form $1,1,4,4,7,7,10,10, \ldots$; indeed, a simple induction shows that $a_{2 k} \equiv a_{2 k-1} \equiv 3 k-2(\bmod p)$ for $k \geq 1$. Since $\operatorname{gcd}(p, 3)=1$ we get that the sequence $a_{n}(\bmod p)$ contains all the residues modulo $p$, a contradiction since only $(p+1) / 2$ residues modulo $p$ are squares. This shows that $m-1$ is a power of 3 .

Let $h, k$ be integers such that $m=3^{k}+1$ and $a_{4}=h^{2}$. We then have $5 \cdot 3^{k}=(h-2)(h+2)$. Since $\operatorname{gcd}(h-2, h+2)=1$, it follows that $h-2$ equals either $1,3^{k}$ or 5 , and $h+2$ equals either $5 \cdot 3^{k}, 5$ or $3^{k}$, respectively. In the first two cases we get $k=0$ and in the last case we get $k=2$. This implies that either $m=2$ or $m=10$.

We now show the converse. Suppose that $m=2$ or $m=10$. Let $t=1$ or $t=3$ so that $m=t^{2}+1$. Let $b_{1}, b_{2}, b_{3}, \ldots$ be a sequence of integers defined by $b_{1}=1, b_{2}=1, b_{3}=2$, and

$$
b_{n}=t b_{n-1}+b_{n-2}, \quad \text { for all } n \geq 4
$$

Clearly, $a_{n}=b_{n}^{2}$ for $n=1,2,3$. Note that if $m=2$ then $a_{4}=9$ and $b_{4}=3$, and if $m=10$ then $a_{4}=49$ and $b_{4}=7$. In both the cases we have $a_{4}=b_{4}^{2}$.

If $n \geq 5$ then we have

$$
b_{n}^{2}+b_{n-3}^{2}=\left(t b_{n-1}+b_{n-2}\right)^{2}+\left(b_{n-1}-t b_{n-2}\right)^{2}=\left(t^{2}+1\right)\left(b_{n-1}^{2}+b_{n-2}^{2}\right)=m\left(b_{n-1}^{2}+b_{n-2}^{2}\right) .
$$

Therefore, it follows by induction that $a_{n}=b_{n}^{2}$ for all $n \geq 1$. This completes the solution.
We present an alternate proof that $m=2$ and $m=10$ are the only possible values of $m$ with the required property.

Note that

$$
\begin{aligned}
& a_{4}=5 m-1, \\
& a_{5}=5 m^{2}+3 m-1, \\
& a_{6}=5 m^{3}+8 m^{2}-2 m-4 .
\end{aligned}
$$

Since $a_{4}$ and $a_{6}$ are squares, so is $a_{4} a_{6}$. We have

$$
4 a_{4} a_{6}=100 m^{4}+140 m^{3}-72 m^{2}-72 m+16 .
$$

Notice that

$$
\begin{aligned}
& \left(10 m^{2}+7 m-7\right)^{2}=100 m^{4}+140 m^{3}-91 m^{2}-98 m+49<4 a_{4} a_{6}, \\
& \left(10 m^{2}+7 m-5\right)^{2}=100 m^{4}+140 m^{3}-51 m^{2}-70 m+25>4 a_{4} a_{6},
\end{aligned}
$$

so we must have

$$
4 a_{4} a_{6}=\left(10 m^{2}+7 m-6\right)^{2}=100 m^{4}+140 m^{3}-71 m^{2}-84 m+36 .
$$

This implies that $m^{2}-12 m+20=0$, so $m=2$ or $m=10$.

Final answer: 1,2