10

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?

PrincessEev
  • 43,815
numberman
  • 101
  • Related: http://math.stackexchange.com/questions/42231/obtaining-irrational-probabilities-from-fair-coins/42236#42236 – Dan Brumleve Jun 02 '11 at 01:45
  • 3
    http://en.wikipedia.org/wiki/Prime_constant – Jonas Meyer Jun 02 '11 at 01:59
  • Reminds me of Liouville's Constant. If I'm not mistaken, wasn't it the first demonstrated example of an irrational number? –  Jun 02 '11 at 02:01
  • 3
    Wiki lists it as a "suspected transcendental" http://en.wikipedia.org/wiki/List_of_numbers#Suspected_transcendentals – Dan Brumleve Jun 02 '11 at 02:02
  • 2
    @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. – André Nicolas Jun 02 '11 at 02:23
  • 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! – amWhy Jun 02 '11 at 03:03
  • 9
    @Dan, every number not known to be algebraic is suspected transcendental. – Gerry Myerson Jun 02 '11 at 03:10
  • @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.) – Jonas Meyer Jun 02 '11 at 03:16
  • 2
    I nominate Jonas to answer this because he found the name. – Dan Brumleve Jun 02 '11 at 03:30
  • 1
    @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... – amWhy Jun 02 '11 at 03:37
  • 4
    Possible duplicate of A binary irrational with bits defined by primes – Klangen Dec 25 '18 at 14:55

1 Answers1

18

Here, since I was "nominated" to answer, is what little I know.

The binary version is called the "prime constant" on some internet sites, but I am not aware of any substantial work on this number.

I initially found it by googling "0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1", which brought me this link to the CRC concise encyclopedia of mathematics with references to OEIS. I could also have entered a similar search on oeis.org. I saw sequence A010051, the characteristic function of the primes. One of the cross references there is sequence A051006, also referenced in the encyclopedia article, which is the decimal expansion of the "prime constant", with that name given. Another Google search with name in hand brings up the Wikipedia article.

Both binary and decimal versions are irrational because the expansions do not repeat, but I can offer no useful comment on the question of transcendentality. As Dan Brumleve notes, a Wikipedia article claims that the binary version is suspected transcendental. As Gerry Myerson notes, "every number not known to be algebraic is suspected transcendental."

Jonas Meyer
  • 53,602
  • thanks all (especially jonas m.)! much appreciated. cheers! numberman – numberman Jun 02 '11 at 05:34
  • The question has now been raised at MO, http://mathoverflow.net/questions/114905/are-these-numbers-irrational-and-or-transcendental where some further references are given (the problem is still open). – Gerry Myerson Nov 29 '12 at 22:18