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Does there exist a number which contains in its digits all of the prime numbers listed in order: $$2.3\,5\,7\,11\,13\ldots\ldots$$

if so, will it be rational or irrational &/or transcendental?

Did
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user125040
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    Non-terminating, non-repeating decimal sounds irrational to me. Transcendental is a tough and interesting question. – John Habert Jan 30 '14 at 19:55
  • Well, you described it, there's no infinite tail of $9$, so it's a valid description of a real number. I don't know if it has any particular properties, however I doubt that this number is rational. – TZakrevskiy Jan 30 '14 at 19:55
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    @JohnHabert non-repeating is not that evident... – TZakrevskiy Jan 30 '14 at 19:56
  • @Newb not everyone is a native speaker here. – TZakrevskiy Jan 30 '14 at 19:57
  • I suppose it's irrational because there are infinitely many prime numbers. A number of this kind is rational if and only if starting from somewhere we have periodic repeating of some set of digits. I guess it's not so easy to formally prove that there wouldn't be repeating. – Poppy Jan 30 '14 at 19:58
  • @TZakrevskiy True enough. Considering that you can't find the sequence 2 3 5 7 11 13 17 inside some of the largest known primes makes it unlikely to repeat. Not a proof though. – John Habert Jan 30 '14 at 20:03
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    Sketch of irrationality: Suppose eventually the digits of the primes form a repeating pattern of length say k. Then for large enough $n$ there are about $k^2$ $n$-digit sequences in this repeating part. But the prime number theorem implies that the number of primes of a given length is unbounded. – Nate Jan 30 '14 at 20:05
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    It's 10 times the Copeland–Erdős constant, which has been proven to be irrational. See here: http://en.wikipedia.org/wiki/Copeland%E2%80%93Erd%C5%91s_constant – benh Jan 30 '14 at 20:10
  • @benh So the remaining question is: is it transcendental? – user125040 Jan 30 '14 at 20:23
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    @user125040: This number is almost surely transcendental, considering the fact that almost all real numbers are transcendental, and this one doesn't have any conceivable reason not to be. At the same time it's probably hard to prove this number transcendental, given that very few numbers have been proved transcendental. – ShreevatsaR Jan 30 '14 at 20:34
  • i'm curious why you think the transcendental question is interesting, @JohnHabert. To me, most such questions with "made up" numbers like this is not interesting at all. – Thomas Andrews Jan 31 '14 at 01:02

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