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I suspect the following statement is true and I can use it in my work if it is true. However I am not a number theorist and I could not prove it myself. I was wondering if this is known to number theorists.

Statement: For each positive real number $\alpha$ there exist a natural number $N$ such that, for every $n \geq N$ each of the intervals $[n^{\alpha},2n^{\alpha}) , [2n^{\alpha},3n^{\alpha}), ...$ contains at least one natural number that has no common factor with $n$.

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    $\begingroup$ You want results on Jacobsthal's function. Some of the simplest explicit results are based on work of Harlan Stevens in a 1977 article. He proves that intervals of length (something like) $2^{2+2e\log k}$ suffice, and this can be improved. $k$ is the number of distinct prime factors of $n$. Check out MathOverflow Question 37679 for more. Gerhard "Still Talking About Jacobsthal's Function" Paseman, 2016.07.31. $\endgroup$
    – Gerhard Paseman
    Commented Jul 31, 2016 at 16:38
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    $\begingroup$ So @GerhardPaseman's mention of question 37679 did not lead to that thread being linked to this one, and maybe he abstained because it was his own thread, but I think it should be a link: (question 37679) Erik Westzynthius's cool upper bound argument: update? $\endgroup$
    – Jeppe Stig Nielsen
    Commented Jul 31, 2016 at 21:39

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This is tightly related to the Jacobsthal function defined to be the smallest integer $j(n)$ such that any segment of $j(n)$ consecutive integers contains an integer co-prime with $n$. Iwaniec ("On the problem of Jacobsthal", Demonstratio Math. 11, 225–231, 1978) proved that $j(n)=O(\log^2(n))$; this implies your statement.

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    $\begingroup$ Is it anyhow related to Cramer's conjecture? The composite numbers between $p_{n}$ and $p_{n+1}$ are coprime with both $p_{n}$ and $p_{n+1}$, and Cramer's conjecture predicts that $p_{n+1}-p_{n}=O(\log^{2} p_{n})$. $\endgroup$
    – Sylvain JULIEN
    Commented Jul 31, 2016 at 17:58
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    $\begingroup$ Sort of not really. The recent paper of Ford Green Konyagin Maynard and Tao observe that most approaches to lower bounds on prime gaps (and thus approaching Cramer's conjecture) take a route involving Jacobsthal's Function, at least implicitly. For Cramer's conjecture, I think not only is Jacobsthal's function needed, but also knowledge of how composite numbers replace the totatives and "fill in the gaps inside the gap". That requires a better understanding and description of how coprimes are distributed than we have at present. Gerhard "It Depends On The Telling" Paseman, 2016.07.31. $\endgroup$
    – Gerhard Paseman
    Commented Aug 1, 2016 at 0:17

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