相关论文: Proof of Bertrand's Postulate for $n\geq 6$
In our effort to find an arithmetically pure proof of the Bertrand postulate, we investigate and solve (using only elementary arithmetical methods) another less usual inequality in positive integers inspired by the classical proof of the…
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…
Inspired by the proof of the Bertrand postulate given by P. Erd\H{o}S, we carefully examine and solve one less usual inequality in positive integers which could help to find an arithmetically pure proof that for every positive integer…
This essay offers a brief biography of Paul Erd\H{o}s and summarizes his approach to mathematics. This is further elucidated by a discussion of Erd\H{o}s' simple proof of Bertrand's Postulate.
In this paper we give an elementary proof for Bertrand's postulate also known as Bertrand-Chebyshev theorem.
An elementary proof of Bertrand's theorem is given by examining the radial orbit equation, without needing to solve complicated equations or integrals.
In 1845, Bertrand conjectured that twice any prime strictly exceeds the next prime. Tchebichef proved Bertrand's postulate in 1850. In 1934, Ishikawa proved a stronger result: the sum of any two consecutive primes strictly exceeds the next…
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…
Alford, Granville, and Pomerance proved that there are infinitely many Carmichael numbers. In the same paper, they ask if a statement analogous to Bertrand's postulate could be proven for Carmichael numbers. In this paper, we answer this…
In this paper, we make some conjectures on prime numbers that are sharper than those found in the current literature. First we describe our studies on Legendre's Conjecture which is still unsolved. Next, we show that Brocard's Conjecture…
This document presents an alternative proof of Sylvester's theorem stating that "the product of $n$ consecutive numbers strictly greater than $n$ is divisible by a prime strictly greater than $n$". In addition, the paper proposes stronger…
We provide empirical evidence for the Erd\H{o}s-Straus conjecture by improving computational bounds to $10^{18}$ and by evaluating the solution-counting function $f(p)$ for this conjecture.
There is no trivial mathematics, there are only trivial mathematicians! A mathematician is trivial if he or she believes that there exists trivial mathematics. Being a non-trivial mathematician myself, I will describe ten different proofs…
In this article we present a class of formulas Fn, n in Nat, that need at least 2^n assumptions to be proved in a normal proof in Natural Deduction for purely implicational minimal propositional logic. In purely implicational classical…
Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…
Bertrand's Postulate ensures existence of prime $p$ between $n$ and $2n$, $n$ an integer $\geq 2$ and the sieve of Eratosthenes, a very simple ancient algorithm, generates all prime numbers up to any given limit. Combining the above two, in…
It is a well-known fact that for any natural number $n$, there always exists a prime in $[n, 2n]$. Our aim in this note is to generalize this result to $[n, kn]$. A lower as well as an upper bound on the number of primes in $[n, kn]$ were…
We present a conclusive answer to Bertrand's paradox, a long standing open issue in the basic physical interpretation of probability. The paradox deals with the existence of mutually inconsistent results when looking for the probability…
Bertrand's Postulate states about the prime distribution for the real numbers. The generalization of Bertrand's Postulate was proved by Das et al. [Arxiv 2018]. In this paper, we have formalized this idea for the Gaussian primes (or the…
An alternative form of Fermats equation[1] is proposed. It represents a portion of the identity that includes three terms of Fermats original equation. This alternative form permits an elementary and compact proof of the first case of…