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Consider the intervals $[0, x)$, $[x, 2x)$, $[2x, 3x)$, ... in $\mathbb{Z}$. Let's call this sequence of intervals $I_1$, $I_2$, $I_3$, ...

Let the function $p(I)$ return the number of primes in an interval $I$.

Does there exist an $x \in \mathbb{Z+}$ such that the sequence $p(I_1)$, $p(I_2)$, $p(I_3)$, ... is non-increasing?

If so, what's the smallest such $x$?

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    $\begingroup$ Since there are arbitrarily long prime-free intervals and infinitely many primes, obviously no. . $\endgroup$
    – fedja
    Commented Aug 23, 2016 at 23:10
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    $\begingroup$ Can't be such an $x$: if there were, $p(I_j)$ would be eventually stationary, say equalling $k$. Obviously $k >0$ (since there are infinitely many primes). However, given $N$, there exists an interval of length $N$ containing no primes, so we obtain a contradiction if $N > 3x$. $\endgroup$
    – David Handelman
    Commented Aug 23, 2016 at 23:16
  • $\begingroup$ I did not see Fedja's comment when I was writing my comment (it took about seven minutes to formulate it, I guess). $\endgroup$
    – David Handelman
    Commented Aug 23, 2016 at 23:18

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