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I am currently studying the following equation:

$p^a(1-p)^b=q^a(1-q)^b$

where $p,q \in (0,1)$, and $a,b \in \mathbb{N}$.

I would like to show that the equation is satisfied if and only if $p=q$.

Is it possible to do this in an exact way? I came across this equation when studying dynamical systems, and I don't have much of a background with these sorts of equations.

(Actually, more precisely, I would like to show that

$\sum_k (p^{a_k}(1-p)^{b_k} - q^{a_k}(1-q)^{b_k}) = 0 \Leftrightarrow p = q$

for $p,q \in (0,1)$ and $a_k,b_k \in \mathbb{N}$.)

Thanks

  • 0
    Sorry, what do you think I should tag it as?2012-05-16
  • 2
    Do you have $a\neq b$? Otherwise $(p,1-p)$ is a solution.2012-05-16

3 Answers 3