Question 197
For a nonnegative integer $n$, define $a_{n}$ to be the positive integer with decimal representation
$$
1 \underbrace{0 \ldots 0}_{n} 2 \underbrace{0 \ldots 0}_{n} 2 \underbrace{0 \ldots 0}_{n} 1
$$
Prove that $a_{n} / 3$ is always the sum of two positive perfect cubes but never the sum of two perfect squares.
$$
1 \underbrace{0 \ldots 0}_{n} 2 \underbrace{0 \ldots 0}_{n} 2 \underbrace{0 \ldots 0}_{n} 1
$$
Prove that $a_{n} / 3$ is always the sum of two positive perfect cubes but never the sum of two perfect squares.