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The problem I'm facing is solving the following equation for $p$ given the constants $a_i$ and $b_i$:

$ \sum_i \frac{a_i}{b_i^p} = 1 $

Is there a general technique that would allow me to find a value for $p$? If not, are there any heuristic approaches that do not require a computer I can use to find a value for $p$?

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    The sum is finite or infinite? the $a_i, b_i$ are real numbers, complex numbers, natural numbers? The exponent $p$ is a prime number, or what?2011-10-25

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Even a very simple example such as ${1\over2^p}+{1\over3^p}=1$ is going to give you an equation that can only be solved by numerical techniques. The example ${1\over2^p}+{1\over32^p}=1$ leads very quickly to the equation $x^5+x-1=0$ which again is going to call for numerical techniques. So I think you are asking for a lot.

EDIT: As DSM points out, my 2nd example is a bad one, that quintic actually factors over the rationals. But there are plenty of $S_5$ quintics of the form $ax^5+bx-1$, just take one and then consider ${b\over2^p}+{a\over32^p}=1$

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    @Gerry Myerson: I always had trouble with minus signs.2011-10-25