Fermat's Last Theorem for n = 3

Statement

There are no positive integers $a$ , $b$ and $c$ with $a^3 + b^3 = c^3$ .

Remarks

A sum of two positive cubes is never a cube. Standard proofs use infinite descent in the ring $\mathbb{Z}[\omega]$ of Eisenstein integers, where $\omega$ is a primitive cube root of unity. This case is part of Fermat's Last Theorem.

History

Fermat mentioned the case $n = 3$ in letters of 1636, 1640 and 1657. Euler published a proof in 1770. Many other proofs followed, for example by Legendre.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

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Proof by infinite descent concept
Fermat's Last Theorem for n = 3 statement

Tag	`07NKG38VEG`
URI	`m2://07NKG38VEG`
Kind	statement, theorem
Writing	W0 stub
MSC	11D25
Status	This copy of the site has no build data. Run `m2 status 07NKG38VEG`.
Uses	Proof by infinite descent

Fermat's Last Theorem for n = 3

Statement

Remarks

History

Reading path

Further reading

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