My professor likes to give our class some questions for fun every once in a while. He posed the following problem in class yesterday, and I've been stuck.

Find the values of $(a,b,c)$ such that $a^{2013}+b^{2013}=c^{2013}$ and $a^2+b^2=c^2$. (The exact wording in the question was that $a,b$ were legs of a right triangle and $c$ was the hypotenuse of the right triangle.) I've thought about trying to find their point of intersection, but I'm not sure how to do that.

Any hints would be appreciated (and would be more preferable than full solutions). Thanks! My professor has stated that this question has nothing to do with our class, so I'm not sure what to tag it. Feel free to change it to something more appropriate.

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    $\begingroup$ Fermat's Last Theorem is quite applicable for the first condition, and will impose some strong conditions. $\endgroup$ – user61527 Sep 28 '13 at 4:41
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    $\begingroup$ If $a,b,c\in \mathbb{R}$, then only $(0,1,1)$ and $ (1, 0, 1)$. $\endgroup$ – Oleg567 Sep 28 '13 at 4:43
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    $\begingroup$ @Oleg567: $0$'s would not be allowed, as the question assumes that $a,b,c$ are sides of a right triangle. $\endgroup$ – Sujaan Kunalan Sep 28 '13 at 4:45

Since we are talking about right triangles, $a$, $b$, and $c$ are positive.

Note that $\left(\frac{a}{c}\right)^2+\left(\frac{b}{c}\right)^2=1$.

Can you argue that $\left(\frac{a}{c}\right)^{2013}+\left(\frac{b}{c}\right)^{2013}\lt 1$?

If we are talking about arbitrary real numbers, there will be some uninteresting solutions.


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