An Inductive Proof of Bertrand's Postulate
Keywords:log (Natural Logarithm) MSC2010: 11N05 (Multiplicative Number Theory, Distribution of Primes)
In this paper, we are going to prove a famous problem concerning the prime numbers called Bertrand's postulate. It states that there is always at least one prime, p between n and 2n, means, there exists n < p < 2n where n > 1. It is not a newer theorem to be proven. It was first conjectured by Joseph Bertrand in 1845. He did not find a proof of this problem but made important numerical evidence for the large values of n. Eventually, it was successfully proven by Pafnuty Chebyshev in 1852. That is why it is also called Bertrand-Chebyshev theorem. Though it does not give very strong idea about the prime distribution like Prime Number Theorem (PNT) does, the beauty of Bertrand's postulate lies on its simple yet elegant definition. Historically, Bertrand's postulate is also very important. After Euclid's proof that there are infinite prime numbers, there was no significant development in the prime number distribution. Peter Dirichlet stated the standard form of Prime Number Theorem (PNT) in 1838 but it was merely a conjecture that time and beyond the scope of proof to the then mathematicians. Bertrand's postulate was a simply stated problem but powerful enough, easy to prove and could lead many more strong assumptions about the prime number distribution. Illustrious Indian mathematician, Srinivasa Ramanujan gave a shorter but elegant proof using the concept of Chebyshev functions of prime, υ(x), Ψ(x)and Gamma function, Γ(x) in 1919 which led to the concept of Ramanujan Prime. Later Paul Erdős published another proof using the concept of Primorial function, p# in 1932. The elegance of our proof lies on not using Gamma function yet finding the better approximations of Chebyshev functions of prime. The proof technique is very similar the way Ramanujan proved it but instead of using the Stirling's approximation to the binomial coefficients, we are proving similar results using well-known proving technique the mathematical induction and they lead to somewhat stronger than Ramanujan's approximation of Chebyshev functions of prime.
GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 85-87
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