On Prime Numbers between kn and (k + 1) n  

作　　者：Wing K. Yu Wing K. Yu(Independent Researcher, Arlington, USA)

机构地区：[1]Independent Researcher, Arlington, USA

出　　处：《Journal of Applied Mathematics and Physics》2023年第11期3712-3734,共23页应用数学与应用物理（英文）

摘　　要：In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always exists at least a prime number p such that kn p ≤ (k +1)n. The Bertrand-Chebyshev’s theorem is a special case of this theorem when k = 1. In the field of prime number distribution, just as the prime number theorem provides the approximate number of prime numbers relative to natural numbers, while the new theory indicates that prime numbers exist in the specific intervals between natural numbers, that is, the new theorem provides the approximate positions of prime numbers among natural numbers.In this paper along with the previous studies on analyzing the binomial coefficients, we will complete the proof of a theorem. The theorem states that for two positive integers n and k, when n ≥ k - 1, there always exists at least a prime number p such that kn p ≤ (k +1)n. The Bertrand-Chebyshev’s theorem is a special case of this theorem when k = 1. In the field of prime number distribution, just as the prime number theorem provides the approximate number of prime numbers relative to natural numbers, while the new theory indicates that prime numbers exist in the specific intervals between natural numbers, that is, the new theorem provides the approximate positions of prime numbers among natural numbers.

关 键 词：Bertrand’s Postulate-Chebyshev’s Theorem The Prime Number Theorem Landau Problems Legendre’s Conjecture Prime Number Distribution 

分 类 号：O17[理学—数学]