How do I find p for equations of the form $\sum \limits_i \frac{a_i}{b_i^p} = 1$

Question

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 $p$?

Answer 1

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 $${b\over2^p}+{a\over32^p}=1$$