Wanted: multiple primes in $(\frac{5n}6,n)$

In an impressive (to me) feat of research, Jose Brox uncovered results of Molsen, Breusch, and Schur regarding the problem. Check https://mathoverflow.net/a/289448 for details.

Edit 2019.03.23:

Even more impressive is the scholarship of Narkiewicz. In his book The Development Of Prime Number Theory, starting on p.116, he briefly outlines the research noted above as well as additional work by Petersen, Gram, Waage, Giordano, Harborth, Kemnitz, and many others on the questions of primes in not very short intervals. This book answers many if not all of the historical questions on prime number theory on MathOverflow. I will recommend it more often.

End Edit 2019.03.23.

Gerhard "It's Easy To Duplicate Answers" Paseman, 2018.10.24.


The following reference will meet your purpose:

P. Dusart, The $k$th prime is greater than $k(\log k+\log\log k−1)$ for $k$ > 2, Math. Comp. 68 (1999), 411–415.

At the end of this paper, the author wrote that "for $x\ge3275$ the interval $[x,x+x/(2\ln^2x)]$ contains at least one prime". This implies your desired result.

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