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Consider $\lfloor 10^n \times 0.731926765612646213686753345587262244668218433356357832021 \rfloor$. For each $n$, reverse the digits. If the number isn't already prime, find the next prime.

For the first 24 values of $n$, the result is prime. This start is based on the record left-truncatable prime 357686312646216567629137.

After that, one must add the following values to get a prime number $(12, 0, 16, 2, 12, 20, 2, 2, 0, 14, 0, 2, 12, 6, 6, 10, 10, 10, 16, 12, 10, 4, 14, 2, 10, 12, 2, 0, 2)$

All values bounded above by 20. If this could go on (it doesn't), it would provide a solution for the finding primes Polymath project. How far can something like this be extended for a given upper bound? For $n$ going from 1 to 100, what constant minimizes the farthest distance to the corresponding 1 to 100 digit prime?

A similar question -- what constants work best without the reverse-digits step? For upper bound 0, $\lfloor 10^n \times .73939133 \rfloor$ could be used to produce primes for $n=1..8$. How far could this be extended with an upper bound of 10?

Is the following provable?
For any $0<k<1$ and any positive integer $d$ there exists an $n$ such that the following values are composite.
$$\lfloor 10^n \times k \rfloor, \lfloor 10^n \times k \rfloor+1, ..., \lfloor 10^n \times k \rfloor +d$$

Added results.
Prime to $n=31$, $\lfloor 10^n \times .19992191983120858559939829 \rfloor$ + $(1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 0, 2, 0, 2, 2, 1, 1, 2)_n$

Prime to $n=43$, $\lfloor 10^n \times .2249916676947644237216638216097398882368887 \rfloor$ + $(0, 1, 3, 2, 2, 2, 1, 1, 0, 1, 0, 3, 2, 1, 3, 3, 1, 0, 2, 1, 0, 3, 3, 0, 1, 1, 0, 1, 1, 0, 2, 0, 0, 3, 1, 3, 1, 0, 1, 1, 3, 1, 0)_n$

Reverse prime to $n=50$, $\lfloor 10^n \times .79946519968197358169933425438229277782582723528486 \rfloor$ + $(0,0,0,2,0,0,2,0,0,0,0,2,2,2,2,2,0,0,0,0,0,0,0,2,2,0,0,2,0,2,2,2,0,2,2,0,2,0,0,2,0,2,2,2,0,2,2,2,0,0)_n$

Reverse prime to $n=64$, $\lfloor 10^n \times .1319788512835399616489432175746323115633163831863127383662373426 \rfloor$ + $(1,0,0,2,2,2,0,2,0,2,2,2,0,0,0,0,0,2,0,2,0,2,2,2,2,2,0,2,0,2,0,0,0,0,0,8,8,6,8,6,2,8,8,0,8,2,0,6,8,2,8,2,8,0,6,6,0,0,6,8,6,8,0,2)_n$

Ed Pegg
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  • For the last question, it's true for all $d$ if $k$ is algorithmically random. If $\lfloor 10^n \cdot k + d \rfloor$ is prime, then we can compress the first $n$ digits of $k$ to a $(n-log(n)+c)$-digit description by referring to it by its index in the set of all primes. – Dan Brumleve Sep 30 '15 at 00:54

1 Answers1

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This is a more detailed version of my comment.

Given $k \in (0,1)$ and $d \in \mathbb{N}$, suppose for all $n \in \mathbb{N}$ there exists a $d_n \in \mathbb{N}$, $0 \le d_n \lt d$, such that $x_n = \lfloor 10^n \cdot k + d_n \rfloor$ is prime.

Lemma: For all $a \gt a_0$, there exists $n$ such that $\lceil \log_{10}(\pi(x_n)) \rceil = a$. In other words, we can always find a prefix of $k$ in base-$10$ of length $n$, add $d_n$ to it, and get a number $x_n$ such that the number of primes less than it has exactly $a$ digits. This can be proved with the prime number theorem.

Now consider $a$ with $a \gt a_0$ and $a = 10^b$ for some $b$, and let $n$ be determined by the lemma. Let $c = \pi(x_n)$. By the prime number theorem:

$\log_{10}(c) = \log_{10}(\frac{x_n}{\log(x_n)}) + O(1) = n - \log_{10}(n) + O(1)$

Also note that:

$\log_{10}(b) = \log_{10}(\log_{10}(n)) + O(1)$

From $b$ we can compute $a$ (the number of digits in $c$), so we can design a prefix-free encoding of $c$ requiring $n - \log_{10}(n) + O(\log_{10}(\log_{10}(n)))$ digits. This leaves room for the constant-sized machinery necessary to compute $x_n$ from $c$ and subtract $d_n$, resulting in the first $n$ digits of $k$, so given any $z$, for sufficiently large $b$ this prefix-free encoding will require fewer than $n-z$ digits. Therefore $k$ is not algorithmically random.

Dan Brumleve
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  • I think I like it. Can you apply this proof to my bound 2 example, and give some range of $n$ where a failure would occur? – Ed Pegg Oct 01 '15 at 16:45
  • This proof also works if you allow $d$ to grow slowly with $n$, for example $d = \log_{10}(n)$. – Dan Brumleve Oct 01 '15 at 17:48
  • If we fix a particular value of $z$ in the argument above, so that $k$ is assumed to have no prefix that can be compressed by more than $z$ digits (Chaitin's constant for example can be characterized this way, $z$ is just the size of a program that can solve the halting problem for programs shorter than $n$ using a length $n$ prefix of it), and assuming we know at least a few of the leading digits of $k$ in order to resolve the $O(1)$ term in the application of the PNT, then it looks like we can find a specific value of $n$ so that all numbers near $\lfloor 10^n \cdot k \rfloor$ are composite. – Dan Brumleve Oct 03 '15 at 17:37