How can I prove that there is no closed form solution to the equation $2^x + 3^x = 10$?
Proof that there is no closed form solution
10
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algebra-precalculus
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12A rigorous proof requires a rigorous definition of "closed form solution". – 2010-11-04
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1Okay. Let say that you can use exponentials, logarithms, digits 0,...,9, variable $x$, $n$th roots, four elementary operations (+ – × ÷) and make compositions and combinations of them. The expression should contain only finitely many characters as written in LaTeX. In particular, symbols $\sum$, $\int$, $\cdots$, $\ldots$ are forbidden. – 2010-11-04
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10@Jaska: in that case you might be interested in reading http://www.jstor.org/stable/2589148 . – 2010-11-04
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0@Qiaochu Yuan: Thanks for that! – 2010-11-04
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0I think that if you could find a function $g(t,u,v)$ such that $x_{0}=g(2,3,10)$, $f(x)=2^{x}+3^{x}-10$, $f(x_{0})=0$, then $2^{x}+3^{x}=10$ would have a closed form. My problem is that I am not able to prove there is no such $g$. – 2010-11-04
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2@Jaska: If you don't have JSTOR access, you can find the paper at http://arxiv.org/abs/math/9805045 – 2010-12-15