Fermat's Last Theorem

After reading Arthur C. Clarke and Frederik Pohl's The Last Theorem,I thought about the problem for a while and I found that the difference between two numbers raised to the power n ( $ c^{n} = (a+b)^{n} - a^{n} $ ) can be expressed as $ \displaystyle \sum_{i=0}^n b \times a^{n-1} \times ( \frac{a+b}{a} )^{i} $.So if it can be proved that $ \displaystyle \sum_{i=0}^n b \times a^{n-1} \times ( \frac{a+b}{a} )^{i} $ cannot be equal to a number raised to the $n$th power then this problem could be solved.

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Easy Math Editor

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$