Are there no positive solutions for $ a^3 + b^3 = c^3$

Question

Fermat's last theorem has no positive solution, for functions of the form of $a^n + b^n = c^n$, where $n > 2$ and $n\in\mathbb{N}$. But I have heard that Fermat wrote in his diary that solutions exist for $a^3 + b^3 = c^3$. Is the thing written in the Diary actually by Fermat, or a rumor spread about.

Answer 1

From Wikipedia (https://fr.wikipedia.org/wiki/Dernier_th%C3%A9or%C3%A8me_de_Fermat#cite_note-4):

"Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere : cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet."

which is rendered as

"It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain."