An Inductive Proof of Bertrand's Postulate

TL;DR

This paper presents a new inductive proof of Bertrand's postulate, establishing prime existence between n and 2n for n > 1, with stronger bounds on the Chebyshev function than previous proofs.

Contribution

The paper introduces an inductive method to prove Bertrand's postulate, providing stronger bounds on the Chebyshev function than earlier proofs by Ramanujan.

Findings

Established stronger bounds for the second Chebyshev function.

Provided an inductive proof approach similar to Ramanujan's method.

Confirmed the existence of primes between n and 2n for n > 1.

Abstract

In this paper, we are going to prove a famous problem concerning prime numbers. Bertrand postulate states that there is always a prime p with n < p < 2n, if n > 1. Bertrand postulate is not a newer one to be proven, in fact, after his assumption and numerical evidence, Chebyshev was the first person who proved it. Subsequently, Ramanujan proved it using properties of Gamma function, and Erd\"os published a simpler proof with the help of Primorial function, p#. Our approach is unique in the sense that we have used mathematical induction for finding the upper and lower bounds for the second Chebyshev function, and they are even stronger than Ramanujan bounds finding using Gamma function. Otherwise, our approach is similar the way Ramanujan proved it.