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Proof of “Fermat’s last theorem”

Let x,y,z,n\in \mathbb{N}^* and n \geq z. Prove that the equation

    \[x^n+y^n=z^n\]

has no solution.

Solution

Without loss of generality , assume that x<y. If x^n+y^n=z^n held , then it would be z^n > y^n thus z^n \geq (y+1)^n. It follows from Bernoulli’s inequality that,

    \begin{align*} z^n &=x^n +y^n\\ &< 2y^n \\ &\leq \left(1 + \frac {n}{z} \right) y^n \\ &\leq \left(1 + \frac {1}{z} \right)^ny^n\\ &< \left(1 + \frac {1}{y} \right)^ny^n \\ &= (y+1)^n \\ &\leq z^n \end{align*}

which is an obscurity. The result follows.

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