# Is the Flajolet-Martin constant irrational? Is it transcendental?

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.

• I didn' know that MO could be used to offer money. Strange, since you can offer up to 500 reputation points through a bounty. – Denis Serre Feb 5 '16 at 16:30
• Jeff is an old-fashioned guy. It also follows the tradition of Donald Knuth. – Deane Yang Feb 5 '16 at 16:41
• Jeffrey, could you please give a reference to Woods&Robbins result. – Al Tal Feb 11 '16 at 10:49
• One can find the decimal digit expansion of this number here: oeis.org/A086744 – Pace Nielsen Feb 11 '16 at 15:30
• D. R. Woods, Elementary problem proposal E 2692, Amer. Math. Monthly 85 (1978), 48. Solution by Robbins. – Jeffrey Shallit Feb 12 '16 at 0:41