I'm reading a book and have encountered a relation which seems to me to be impossible to prove, I would like to be sure if this is the case. The author gives a probability function as $$p_n = \frac{e^{-c_1 n - c_2/n}}{Z},$$ where $c_1$ and $c_2$ are constants and Z is a normalization factor and $n \geq 3$. Then by considering $\langle n \rangle = 6$ and defining $\alpha$ (the second moment) as $\alpha = \sum_{n = 3}^{\infty} p_n (n - 6)^2$, the author claims one can show that

\begin{equation} \alpha + p_6 = 1, \quad \quad \quad 0.66 < p_6 < 1, \end{equation} \begin{equation} \alpha p_6^2 = 1 / 2 \pi, \quad \quad \quad 0.34 < p_6 < 0.66. \end{equation}

How is such a thing possible in the first place as these relations are not even dependent on $c_1$ and $c_2$?