Talk:Copeland–Erdős constant

Analogue

If the constant created by concatenating all primes of the form $4n+1$ is rational, there exists an integer $s$ such that there is no prime of the form $4n+1$ of $s$ digits. (Hardy and Wright, p. 113)

0.5131729374153...

218.133.184.93 17:08, 6 July 2007 (UTC)

Well, your reference is mistaken, as I have Hardy and Wright (5th edition) p 113 open in front of me, and it says no such thing. What it does give is two proofs that the Copeland-Erdős constant is irrational. One is the proof based on Dirichlet's theorem, as referenced in the main article; the other uses the fact that if there is a prime between N and 10N for every N>=1 (which in turn follows immediately from Bertrand's postulate) to show that the Copeland-Erdős constant cannot be a recurring decimal. Gandalf61 18:34, 6 July 2007 (UTC)

Well,you got that book alright then you have to understand what I'm saying is true from the fact that they're using Bertrand's postulate for the second proof. 218.133.184.93 18:44, 6 July 2007 (UTC)

Logically, it is trivially true - since the constant 0.5131729374153... is not rational, then if we suppose it is rational, any consequence at all will logically follow from this. However, it is not a useful statement to insert into the article because:

1. The reference you give from Hardy and Wright does not discuss this actual number.
2. The way you have phrased the statement implies that this number might be rational, whereas we know that it is not rational (from the general proof based on Dirichlet's theorem that is given earlier in the article).
3. You can use Bertrand's postulate to prove that the Copeland-Erdős constant is irrational, but this proof does not generalise to the other constants - Bertrand's postulate does not say that there must be a prime of a particular form (say 4k+1) between n and 2n.
4. The irrationality of 0.5131729374153... is a special case of the general fact that all of these constants, created from primes in a given arithmetic sequence, are irrational. This fact is already discussed and proved in the article - an alternative proof of a special case (even if you clarified how the alternative proof might work) adds nothing of value.

Please stop re-inserting this paragraph. If you wish to contribute to Wikipedia, find a more useful and less disruptive way to do that. Gandalf61 12:24, 8 July 2007 (UTC)

Note that User:218.133.184.93 is in Category:Suspected Wikipedia sockpuppets of WAREL, like the almost identical IP User:218.133.184.53. Warel has a long history of disruption in math articles. PrimeHunter 12:43, 8 July 2007 (UTC)

The statement that there exist primes of the form

$k10^{m}+1$

for all positive integers $k$ and $m$ is not true. 218.133.184.93 14:06, 4 August 2007 (UTC)

You are right. Well spotted. If k and m are both given then it's a constant which is often composite. It should only say for all m. I have fixed it. PrimeHunter 15:28, 4 August 2007 (UTC)

Hey, I wonder if this anonymous editor's (you know who I mean) problem with the irrationality argument is due to Hardy and Wright's presentation of the proof. There is no proof of Dirchlet in HW, so the argument for the irrationality begins "Let us assume that any arithmetical progression...". It is pointed out below the proof that Dirichlet is assumed in this proof, so the "assume" is really there to make it clear to the reader that an assumption is being made that has not, and will not, be proved in the book, even though it is true. This is why there are two proofs of this irrationality: the second proof does not rely on Dirichlet. Just a thought. Doctormatt 23:22, 13 August 2007 (UTC)

I intend to revert to a clean copy (without the m+2 and m+1 I added) in about 22 hours, if no one else does it. — Arthur Rubin | (talk) 00:53, 14 August 2007 (UTC)

Twin prime version

Can anyone prove the constant created by concatenating prime p that p+2 is also a prime

0.35111729...

is irrational?

Assume that there are infinitely many twin primes.

Thank you for your help. Motomuku (talk) 10:15, 13 June 2009 (UTC)

I don't think this follows from current knowledge about twin primes even if we assume there are infinitely many. Wikipedia:Reference desk/Mathematics may be a better place to ask. PrimeHunter (talk) 11:26, 13 June 2009 (UTC)

It is not even known if it is a number with infinite digits (so it can theoretically be a rational). --Infovarius (talk) 11:33, 10 January 2018 (UTC)

Transcendental

Is there any information about transcedentality of the constant? --Infovarius (talk) 11:33, 10 January 2018 (UTC)

1 a prime?

Copeland and Erdős apparently considered 1 a prime number, and they defined the constant as 0.12357111317... Of course, this doesn't change any of the number's important properties. Cocaineninja (talk) 18:17, 6 October 2019 (UTC)

Generalisation

Copeland and Erdős's proof that the constant is normal uses only the fact that $p_{n}$ is strictly increasing and $\pi (n)=n^{1-o(1)}$, so it works for any sequence with similar properties. Cocaineninja (talk) 12:41, 7 October 2019 (UTC)