Problem 1409

To show that if an arithmetic sequence contains one perfect square, then it contains infinitely many perfect squares, we can use the properties of arithmetic sequences and modular arithmetic.

1. Define the Arithmetic Sequence:
Let the arithmetic sequence be given by $a_n = a + nd$, where $a$ is the first term and $d$ is the common difference.

2. Assume the Existence of a Perfect Square:
Suppose there exists an integer $n_0$ such that $a_{n_0}$ is a perfect square. That is, $a + n_0 d = k^2$ for some integer $k$.

3. Consider the General Form of Perfect Squares in the Sequence:
We need to show that there are infinitely many integers $n$ such that $a + nd$ is a perfect square. Consider the term $a + (n_0 + td)d$ for some integer $t$. This term can be written as:
$$a + (n_0 + td)d = a + n_0 d + tdd = k^2 + tdd$$

4. Express the Term as a Perfect Square:
We need to find $t$ such that $k^2 + tdd$ is a perfect square. Let $k_t$ be an integer such that:
$$k_t^2 = k^2 + tdd$$
This implies:
$$k_t^2 - k^2 = tdd$$
$$(k_t - k)(k_t + k) = tdd$$

5. **Solve for $t$:**
Since $d$ is fixed, we need to find $t$ such that $tdd$ can be expressed as the product of two integers $(k_t - k)$ and $(k_t + k)$. Notice that if we choose $t$ to be a multiple of $d$, say $t = md$ for some integer $m$, then:
$$tdd = mdd^2$$
$$(k_t - k)(k_t + k) = mdd^2$$

6. Conclusion:
By choosing $t$ appropriately, we can always find $k_t$ such that $k_t^2 = k^2 + tdd$ is a perfect square. Since there are infinitely many choices for $t$, there are infinitely many perfect squares in the sequence.

$\blacksquare$