Given $a,b,c \in \mathbb{N}$ which satisfy the following conditions:
$a^3 + b^3 = c^2$
$ a \neq b$
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EDIT, Will Jagy: The conjecture is that, for a given $c,$ there are at most two distinct pairs $(a,b)$ with $1 \leq a < b$ and $a^3 + b^3 = c^2.$ Note the example $77976^2=1026^3+1710^3=228^3+1824^3$ that Gerry found with exactly two pairs.
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There are maximum $2$ possible pairs of $(a, b)$ which satisfy this conditions.
I have verified it for $a, b \leq 200\,000$.
Note: the situation when $a$ and $b$ just take each others value is discarded.