Existence of primes in the interval $[15x,16x]$ -- An entirely elementary proof --

Abstract

In this paper, we give a short and entirely elementary proof of the proposition ``For any positive integer $N$, there exists a real number $L$ such that for any real number $x\geqq L$, there are at least $N$ primes in the interval $[kx,(k+1)x]$'' for $k\leqq 15$. Our proof is based on the idea of the proof by Erdös for $k=1$ and its improvement by Hitotsumatsu and by Sainose for $k=2$. In the case of $k=3$ and $k=4$, the method is very similar to the case of $k=2$, however, in the case of $k\geqq 5$, we need new idea to complete the proof.

15 pages. This manuscript is a more recent version, correcting some minor errors from the published version