For a nonnegative integer
, define
to be the positive integer with decimal representation
Prove that
is always the sum of two positive perfect cubes but never the sum of two perfect squares.
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For a nonnegative integer $n$, define $a_n$ to be the positive integer with decimal representation
$$1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.}$$
Prove that $\frac{a_n}{3}$ is always the sum of two positive perfect cubes but never the sum of two perfect squares.
Školjka