How to check if a number can be represented as difference of a cube and square ?
For eg. $18 = 27 - 9$. Hence $18$ can be represented as difference of a cube and square.
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This can be computed by the so-called Mordell curve, the elliptic curve $$ y^2=x^3+k. $$ For a reference see the notes by K. Conrad, or the references here at MSE: