2008/11/26 by Shiva Kintali, Kintali, Shiva
Arts and Humanities · Mathematics · #Analytic Number Theory Research #Chebyshev filter #Combinatorics #Computer science #Conjecture #Discrete mathematics #History and Theory of Mathematics #History of Science and Medicine #Integer (computer science) #Legendre polynomials #Mathematical analysis #Mathematics #Prime (order theory) #Prime number #Prime number theorem
paper · pdf · doi:10.48550/arxiv.0811.4451
openalex publication_date 2008/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Legendre's conjecture states that there is a prime number between n2 and (n+1)2 for every positive integer n. We consider the following question : for all integer n>1 and a fixed integer k<=n does there exist a prime number such that kn < p < (k+1)n ? Bertrand-Chebyshev theorem answers this question affirmatively for k=1. A positive answer for k=n would prove Legendre's conjecture. In this paper, we show that one can determine explicitly a number N(k) such that for all n >= N(k), there is at least one prime between kn and (k+1)n. Our proof is based on Erdos's proof of Bertrand-Chebyshev theorem and uses elementary combinatorial techniques without appealing to the prime number theorem.