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Recently while coming up with an example for a paper I'm writing I find myself wanting something to say about how 'awful' the first positive root of the equation $ 4\cdot2^{2p}\cdot3^p-4\cdot3^{2p}-7\cdot2^{3p}+8\cdot2^{2p}+8\cdot3^p-4=0 $ is. Numerically I know it's about 1.576, but I suspect that it's no only irrational but cannot be solved for using elementary functions, e.g. $\log_2(3+\sqrt{5})$.

I've not much / any background in number theory of things like this and am kind of stumped. Are there any simple arguments in this direction?

Note: This is the simplest of the awful equations I could come up with. I'm also aware that it has a root at $p=2$

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    @Gerry Myerson. Thanks for spotting that. I'll have a further bash at it.2011-07-01

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I suspect there aren't any simple arguments for showing that one or all of the solutions to an equation of this kind is not expressible using elementary functions. A 1999 paper of Tim Chow reviews one reasonable way to formally define "elementary" in this context, and points out that proving anything to be non-elementary seems to be very difficult.

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    Thanks. I thought as much, but like I said I have very little experience in this. Thanks for the link to the paper.2011-06-28