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
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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
2 Answers
By the way, your equation can be written in form
$$H^{(x)}_3=11$$
where H is the generalized harmonic number: http://www.wolframalpha.com/input/?i=HarmonicNumber[3%2C+x]%3D%3D11
So to find x you should investigate the inverse function of generalized harmonic number.
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0$H_{3}^{(-x)}=11$ – 2017-09-10
A closed-form solution is a solution wich can be expressed as a closed-form expression.
A mathematical expression is a closed-form expression iff it contains only finite numbers of only constants, functions, operations and/or variables.
Sensefully, all the constants, functions and operations in a given closed-form expression should be from allowed sets.
The following answer is only for closed-form solutions that are expressions of elementary functions.
According to Liouville and Ritt, the elementary functions can be represented in a finite number of steps by performing only algebraic operations and / or taking exponentials and / or logarithms.
$2^x+3^x=10$ is a transcendental equation: $e^{x\ln(2)}+e^{x\ln(3)}=10$. The left-hand side of this equation is the functional term of an elementary function. Because $\ln(2)$ and $\ln(3)$ are linearly independent over $\mathbb{Q}$, one can prove by simple arguments that $x$ cannot be isolated only on one side of the equation by only applying only elementary functions and/or algebraic operations to the equation. Moreover, by applying the theorem of [Ritt 1925], one can prove the function $2^x+3^x$ cannot have a local inverse that is an elementary function.
But the non-existence of a local elementary or local closed-form inverse is not an exclusion criterion for the existence of an elementary respective closed-form solution.
The question of solvability of equations by elementary functions is treated in [Rosenlicht 1969].