Why chord and tangent method does not give further points on Fermat's curve?

Question

(0,1) and (1,0) are two rational points on x^3 + y^3= 1. But why doesn't chord and tangent method yield any further points on the curve?

Answer 1

There are no additional rational points on the curve, since the equation $x^3+y^3=z^3$ has no non-trivial solutions in integers. This is the case $n=3$ of Fermat's Last Theorem, probably first proved by Euler.