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Consider the number with binary or decimal expansion

$$0.011010100010100010100...$$

that is, the $n$'th entry is $1$ iff $n$ is prime and zero else. This number is clearly irrational. Is it known whether it is transcendental?

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    Related: http://math.stackexchange.com/questions/42231/obtaining-irrational-probabilities-from-fair-coins/42236#422362011-06-02
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    http://en.wikipedia.org/wiki/Prime_constant2011-06-02
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    Reminds me of [Liouville's Constant](http://en.wikipedia.org/wiki/Liouville_number). If I'm not mistaken, wasn't it the first demonstrated example of an irrational number?2011-06-02
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    Wiki lists it as a "suspected transcendental" http://en.wikipedia.org/wiki/List_of_numbers#Suspected_transcendentals2011-06-02
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    @Graham Enos: You mean of a transcendental number, in which case the answer is yes. By tradition, the first incommensurability proof involved $\sqrt{2}$, though some have argued it might have been the so-called *golden number*.2011-06-02
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    Yes Liouville's Constant was the first demonstrated example of a transcendental number. So I take it that, the proposed number can be taken taken to be the binary decimal expansion of the "Prime Constant"? Interesting links, all!2011-06-02
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    @Dan, every number not known to be algebraic is suspected transcendental.2011-06-02
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    @amWhy: To be a pedant, you mean "binary", not "binary decimal". The *decimal* number 0.011010100010100010100... is also asked about in the question, but isn't mentioned in the links above. (The phrasing in the question is confusing, because it refers to "the" number but really seems to be asking about two different numbers.)2011-06-02
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    I nominate Jonas to answer this because he found the name.2011-06-02
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    @Jonas: you're absolutely correct...I was thinking of both version of the prime constant...forgive me? ;-) And yes, I second the nomination for Jonas to answer...2011-06-02
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    Possible duplicate of [A binary irrational with bits defined by primes](https://math.stackexchange.com/questions/1521820/a-binary-irrational-with-bits-defined-by-primes)2018-12-25

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