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Proof of “Fermat’s last theorem”

By Tolaso in Uncategorized on October 5, 2019.

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,

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