Is the following number transcendental? $$0.23571113171923293137\dots$$(Obtained by writing prime numbers consecutively from left to right, in the decimal expansion)

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    $\begingroup$ I really can't imagine that number being the root of any rational number. Also, I can't imagine it is a number of any mathematical significance, since what number you get depends on what base you're in. What brought you to this question? $\endgroup$ – Arthur Sep 16 '12 at 12:11
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    $\begingroup$ @Arthur: You mean "a root of any rational polynomial", right? As for mathematical significance, it is not obviously less worthy of consideration than the Champernowne constant or Liouville's number. $\endgroup$ – Henning Makholm Sep 16 '12 at 12:13
  • $\begingroup$ @Clive: That's an answer! $\endgroup$ – Henning Makholm Sep 16 '12 at 12:17
  • $\begingroup$ @HenningMakholm: If you say so ;) $\endgroup$ – Clive Newstead Sep 16 '12 at 12:19
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    $\begingroup$ Classify mathematical objects in "significant" and "not significant" is a good way to limit yourself in mathematics. If everyone since the beginning of history would merely make significant math, we would never be at this level of mathematics we have today. $\endgroup$ – Integral Sep 16 '12 at 15:37

This number is called the Copeland–Erdős constant, and is known to be irrational and normal. I believe its transcendence or otherwise is an open problem. This source claims that it has been proved to be transcendental, but the paper they refer to is the one in which it was proved to be normal and so I think the source is mistaken.

For now, the knowledge that it is almost surely transcendental will have to suffice!


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