Variation on Fermat's Last Theorem

I've already heard about how Fermat stated that no three positive integers $a$ , $b$ and $c$ can satisfy the equation $a^n + b^n = c^n$ for any value of $n>2$ . I think Andrew Wiles proved that this conjecture was true.

Now, I'm wondering whether any three positive integers $a$ , $b$ and $c$ can satisfy the equation $\frac {1}{a^n}$ + $\frac {1}{b^n}$ = $\frac {1}{c^n}$ for any value of $n>2$ .

Note: If this question was already asked please show me the link of that discussion so I can look over that.

If this is possible tell me why it is possible.

If this is impossible tell me why it is impossible.

#NumberTheory

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Comments

You can write the equation as $(bc)^n+(ac)^n=(ab)^n$ , so, it's impossible

Otto Bretscher - 5 years, 5 months ago

Thank you very much!

Ananth Jayadev - 5 years, 5 months ago

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}`	$\boxed{123}$