TITLE:
On Prime Numbers between kn and (k + 1) n
AUTHORS:
Wing K. Yu
KEYWORDS:
Bertrand’s Postulate-Chebyshev’s Theorem, The Prime Number Theorem, Landau Problems, Legendre’s Conjecture, Prime Number Distribution
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.11 No.11,
November
30,
2023
ABSTRACT: 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.