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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.

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

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