Ramanujan's Proof of Bertrand's Postulate

Bertrand's postulate

Bertrand’s postulate is an early result on the distribution of prime numbers: For every positive integer n, there exists a prime number that lies strictly between n and 2n. The proof is ported from John Harrison’s formalisation in HOL Light [1]. It proceeds by first showing that the property is true for all n greater than or equal to 600 and then showing that it also holds for all n below 600 b...

A proof of Bertrand's postulate

We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, subsuming, as a corollary, Bertrand’s postulate. Even if Chebyshev’s result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions ψ and θ still play a central role in the modern development...

Arithmetic in Metamath, Case Study: Bertrand's Postulate

Unlike some other formal systems, the proof system Metamath has no built-in concept of “decimal number” in the sense that arbitrary digit strings are not recognized by the system without prior definition. We present a system of theorems and definitions and an algorithm to apply these as basic operations to perform arithmetic calculations with a number of steps proportional to an arbitrary-preci...

Ramanujan Primes and Bertrand's Postulate

A Combinatorial Proof of an Identity from Ramanujan's Lost Notebook

In this note, we present a new combinatorial proof of an identity originally stated in Ramanujan’s lost notebook by keeping track of an extra statistic in Yee’s bijection for a special case of this identity.