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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.

Air Mike
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sqrt
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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:

Are all Mordell equations $y^2=x^3+k$, for any integer $k$, solvable

Solutions to $y^2 = x^3 + k$?

Dietrich Burde
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    This is the first time the relationship has occurred to me, two squares minus a cube give everything: http://zakuski.utsa.edu/~jagy/Elkies_Kap.pdf – Will Jagy Aug 16 '20 at 17:47
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    @WillJagy Ah, very interesting. I wasn't aware of it, too. Perhaps we'll see this soon here as a new question. – Dietrich Burde Aug 16 '20 at 18:12
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    Interesting material. More in http://zakuski.utsa.edu/~jagy/papers/Experimental_1995.pdf where we informed Vaughan early enough for him to put our main example in the second edition of his book on the Hardy Littlewood method. We had not known about this, Kevin Ford told us that we had disproved an existing conjecture which was recorded in Vaughan's first edition. – Will Jagy Aug 16 '20 at 20:15