Prove that x^{n} + y^{n} = z^{n} is impossible for x, y, z, n positive integers, with x \lt n, y \lt n, n \gt 2.

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Proof

Without loss of generality we may assume x \gt y. Then

y^{n} = z^{n} - x^{n} = (z - x)(z^{n-1}+z^{n-2}x + \ldots + x^{n-1}).

From here,

y^{n} \gt 1 \cdot nx^{n-1} \gt y^{n}.

A contradiction.