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Here is the sequence formed by concatenating the sequence of prime numbers in base $10$:

$2,3,5,7,\color{red}{1},\color{red}{1},\color{blue}{1},\color{blue}{3},\color{orange}{1},\color{orange}{7},\color{brown}{1},\color{brown}{9},\color{green}{2},\color{green}{3},\dots$

(This is A033308, the sequence of digits in the Copeland-Erdos constant, which is a normal number.)

Let $P(n)$ be the proportion of odd terms among the first $n$ terms.

Here is the graph of $P(n)$ against $n$, from $n=1$ to $n=1000$:

enter image description here

Among the first $1000$ terms, the maximum value of $P(n)$ is $P(12)=\frac{11}{12}\approx 0.91667$.

Now consider the following claim:

The absolute maximum value of $P(n)$ is $\frac{11}{12}$.

No one would seriously doubt that the claim is true. Here is some data about the proportion of odd terms in the sequence. For the claim to be false, there would have to be an overwhelming bias toward odd digits deep into the unknown stretches of the prime number sequence, which seems preposterous.

But can the claim be proved?

(If the claim cannot be proved with current knowledge, then this would top my list of intuitively true claims that are unproven; higher on the list than the claim that $\pi+e$ is irrational, and the claim that $\pi$ is normal.)

Dan
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  • This is like the opposite of concatenating. I can't remember the CS word for it, but you're breaking up the digit sequence into its individual elements – FShrike Jul 24 '23 at 12:31
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    I would seriously doubt the claim is true, anyway. Not because I have reason to believe it's false, but because I have no reason to believe that it is true. – FShrike Jul 24 '23 at 12:32
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    @FShrike Fair enough. But I think the claim in the question is similar to the claim that "In the decimal expansion of $\pi$, it is not the case that, after a certain point, the rest of the terms are all $0$ and $1$". I can't prove it's true, but I have a very strong intuition about it. Anyway, it's just my opinion. – Dan Jul 24 '23 at 12:48
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    Concerning the other conjectures : That $e+\pi$ is irrational , even transcendental , has a strong evidence (although it is currently unproven). That $\pi$ is normal , does not have any real evidence. We are extremely far away from proving this. We should first be able to prove that $7$ appears infinite many often in $\pi$. I guess we are even screwed with this. – Peter Jul 24 '23 at 13:43
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    As a general matter, questions that mix broad principles, like primality, with questions specific to base $10$ representations are very hard to address. Naively, I'd expect the ratio to be $\frac 12$, with small number bias attributable to the fact that we know the final digit to be odd. But I wouldn't expect anything much to be provable. – lulu Jul 24 '23 at 13:43
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    Another bias towards odd digits comes from Benford's law. – Peter Jul 24 '23 at 13:48
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    This weird observation is surely worth exploring in other bases and with other partitions of the set of digits. – Ethan Bolker Jul 24 '23 at 14:00
  • I wouldn't be too suprised if it could be proven. It could be possible to give rough upper/lower bound with Dirichlet's theorem on arithmetic progressions. I'm probably not good enough to prove it, but the formulas here could be useful: https://en.wikipedia.org/wiki/Prime_number_theorem#Prime_number_theorem_for_arithmetic_progressions – Michael Stocker Jul 24 '23 at 14:29
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    I would be VERY surprised if it could be proven. The conjecture is true for the first $4$ million digits. The ratio is at least $0.57$ for $n>2$ and at most $0.7$ for $n>1061$ upto this point. The final ratio is about $0.581$ after the primes upto $10^7$ are concattenated. – Peter Jul 24 '23 at 15:33

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