Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm.

In the Flajolet-Martin paper, they define a certain constant $\varphi = 2^{-1/2} e^\gamma \alpha^{-1}$, where $\alpha = \prod_{n \geq 1} ({{2n}\over{2n+1}})^{(-1)^{t(n)}}$ and $t(n)$ is the ubiquitous Thue-Morse sequence, which counts the parity of number of $1$'s in the binary expansion of $n$.

Is $\alpha$ irrational? Is it transcendental? I offer $50 for a proof or disproof of either of these.

As an aside, I mention that the closely related infinite product $\prod_{n \geq 0} ({{2n+1}\over{2n+2}})^{(-1)^{t(n)}}$ is known to be equal to $2^{-1/2}$. This was originally found by Woods and Robbins.