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Let $k \geq 3$ be a (large enough) integer, let $x \in \mathbb{R}$, and set $I_x := [x, x + \log^k x]$.

Some believe that for $x$ large enough there exists a prime $n \in I_x$. Equivalently, there exists (for large enough $x$) an $n \in I_x$ with $\omega(n) = 1$ where $\omega(n)$ is the number of distinct prime factors of $n$. This seems to be out of reach for the moment.

What (upper) bound can be proved (even assuming big conjetures like GRH or ABC) on $$\min \ \{\omega(n) \ | \ n \in I_x \}$$ for large enough values of $x$?

The (very weak) bound that I managed to obtain is: $C(k)\frac{\log x}{\log \log x}$ where $C(k)$ is approximately (an absolute constant times) $1/k$. This improves upon the trivial upper bound $O(\frac{\log x}{\log \log x})$. I am therefore eager to know if a bound of the form $o(\frac{\log x}{\log \log x})$ can be established.

For (substantially) longer intervals, the Erdos-Kac theorem holds, so much better bounds are available.

I am not interested in results about almost all values of $x$, or about averaging over $x$.

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  • $\begingroup$ What prevents you from proving results like log log x for the minimum? I assume the width of the interval is $(\log x)^k $ which is greater than $2^k$? There can't be that many occurrences of small prime factors, can there? Gerhard "Is Also Interested In This" Paseman, 2017.07.11. $\endgroup$
    – Gerhard Paseman
    Commented Jul 11, 2017 at 15:31
  • $\begingroup$ @GerhardPaseman I took into account the occurences of small primes, and from that I could not arrive at $\log\log x$ but only at that weak bound mentioned in the question. $\endgroup$
    – Pablo
    Commented Jul 11, 2017 at 15:34
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    $\begingroup$ MathOverflow user Joni Teravainen (apologies for poor typesetting and no umlauts) cites the Mikawa paper and says that he can almost always find bounds of 2 when the exponent on log is 3.51, and of 3 for exponent 1+ epsilon. He has an ArXiv print from 2015 on this (1510.06005). Not quite what the poster wants though. Gerhard "Is Making A Gender Presumption" Paseman, 2017.07.11 $\endgroup$
    – Gerhard Paseman
    Commented Jul 12, 2017 at 5:55
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    $\begingroup$ Which is why your comment surprised me Eric. However, it may be the same exponent will work for him. Do you know how to extend Mikawa's (or others) work from almost all x to all x? Gerhard "Occasionally Wonders About Others Comments" Paseman, 2017.07.12. $\endgroup$
    – Gerhard Paseman
    Commented Jul 12, 2017 at 19:01
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    $\begingroup$ The scenario that needs eliminating is when every number in $[x, x+\log^k x]$ is the product of about $\frac{1}{100k} \frac{\log x}{\log\log x}$ primes of size comparable to $\log^{100 k} x$, times some smaller primes (up to $\log^k x$ in size, but a typical number in the interval would only have about $O(k \log\log x)$ such factors), with none of these medium sized primes being used more than once. I don't know of any technique (even assuming ABC etc.) that would eliminate this scenario. $\endgroup$
    – Terry Tao
    Commented Nov 21, 2017 at 17:44

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