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Paint the number line from each prime number to its closer prime neighbor; in the case of a tie, choose the lower neighbor.

enter image description here

What is the limiting proportion of the positive number line that is painted?

That is, what is the limit, as $n\to\infty$, of the proportion of the number line from $0$ to $n$ that is painted?

Here is what I have found:

enter image description here

If, instead, we have uniformly random real numbers in an interval, then as the number of random numbers approaches infinity, the limiting proportion is $7/18$ (this is the "Birds on a Wire" problem, which inspired this question).

Dan
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  • I doubt if this question can be answered with the current knowledge of primes. We don't even know an upper bound on the gap between two consecutive primes; this question requires information about the gap between three consecutive primes. – Nilotpal Sinha Dec 27 '24 at 10:35
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    On second thought, I think for primes too, it should converge to the same limit $7/18$ as in the birds on a wire problem. Give me some time to draft an answer. – Nilotpal Sinha Dec 27 '24 at 10:44
  • Is the graph in your post the one for primes or the one of for the uniform distribution case? – Nilotpal Sinha Dec 27 '24 at 10:53
  • @NilotpalSinha The graph is for primes. – Dan Dec 27 '24 at 10:54

1 Answers1

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Let $p_n$ be the $n$-th prime. Consider the numbers from $p_1$ to $p_n$. Divide each number by $p_n$ to get the sequence $\displaystyle \frac{2}{p_n}, \frac{3}{p_n}, \cdots, 1$. This sequence approaches uniformly distribution in $(0,1)$ as $n \to \infty$. Hence in the limiting case, the proportion of number line colored in the case for primes will converge to that of the uniform distribution case i.e. $\displaystyle \frac{7}{18}$.

Nilotpal Sinha
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