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$f(k) = \sum_{r=1}^{n} r^k$. Find an integer $x$ that solves the equation $f(x) = \bigl(f(1)\bigr)^2$.

Problem credit: http://cotpi.com/p/2/

I understand why $x = 3$ is a solution. $1^3 + 2^3 + \dots + n^3 = \left(\frac{n(n + 1)}{2}\right)^2 = (1 + 2 + \dots + n)^2$. But how can we prove that there is no other solution?

Will we have other solutions if $x$ is real? What if $x$ is complex?

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    (Note: the title had its variables wrong in the sum limit and the expressions. I've fixed it to match the question.)2012-05-24

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