2025/03/08 by Aoki, Hiroki, Higa, Riku, Sugawara, Ryosei
#11A41 #11N05 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2503.06069
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.