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I am currently dealing with the question of which numbers can be expressed as a sum of two rational cubes ($n=x^3+y^3$, $x,y\in\mathbb{Q}$). I've already learned that I can understand the equation $n=x^3+y^3$ as an elliptic curve (given by Weierstraß-form $y^2=x^3+Ax+B$) and try to find rational points on this curve.

My question is: How can the equations $n=x^3+y^3$ and $y^2=x^3+Ax+B$ be transformed into each other?

And in what way does projective geometry play a role here?

P.S.: I've started to read the Elliptic Curve Handbook by Conwell(1999) and Taxicabs and Sums of Two Cubes by Silverman (1993), but it feels like I need some rough intuitive introduction before being able to understand it completely. Thanks!

känguri
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    Does this answer your question? elliptic curve ${X^3+Y^3=AZ^3}$. It is the elliptic curve with $[a_1,a_2,a_3,a_4,a_6]=[0, 0, -9n, 0, -27n^2]$. – Dietrich Burde Sep 26 '22 at 13:58
  • This is just a fun fact related to your question, which I recently read, but maybe you and others are interested: An answer here (https://math.stackexchange.com/q/1464174) states, that Legendre believed, that $6$ isn't the sum of rational cubes, which turned out wrong as $6=\left({17\over21}\right)^3+\left({37\over21}\right)^3$. – Samuel Adrian Antz Sep 26 '22 at 14:03
  • Sorry, the duplicate answers it more precisely. It is a Selmer curve with short Weierstrass form $y^2=x^3-432n^2$. We have $a=b=1$ there, and $c=n$, with Proposition $1.4.1$ in Conwell's text you have cited. – Dietrich Burde Sep 26 '22 at 14:07
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    Thank you both - indeed, Prop. 141 and Cor. 1.4.2 in Conwells Book is what I searched for - I must have been unconcentrated. Thanks @DietrichBurde :) – känguri Sep 26 '22 at 15:00

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