Is the following number transcendental? $$0.23571113171923293137\dots$$(Obtained by writing prime numbers consecutively from left to right, in the decimal expansion)
Is $0.23571113171923293137\dots$ transcendental?
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$\begingroup$
number-theory
transcendental-numbers
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1I 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? – 2012-09-16
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9@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](http://en.wikipedia.org/wiki/Champernowne_constant) or [Liouville's number](http://mathworld.wolfram.com/LiouvillesConstant.html). – 2012-09-16
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0@Clive: That's an answer! – 2012-09-16
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0@HenningMakholm: If you say so ;) – 2012-09-16
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0@HenningMakholm Yes, yes I do. As for the other constants, I wasn't aware they had been studied at all, except for the trancendentality of Liouville's number, of which I didn't know the historical significance. – 2012-09-16
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4Classify 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. – 2012-09-16
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0@Integral Maybe the current level of mathematics is quite honorable. – 2012-09-16
1 Answers
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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!