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 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.