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

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 Erd\"{o}s 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.