scholarly journals On Primes of the Form 2x-q (where q is a prime less than or equal to x) and the Product of the Distinct Prime Divisors of an Integer (Revised): A Function Approach to Proving the Goldbach Conjecture by Mathematical Induction

2021 ◽  
Author(s):  
Chukwunyere Kamalu
Keyword(s):  
Upper Bound ◽  
Multiplicative Function ◽  
Mathematical Induction ◽  
Function Approach ◽  
Geometric Series ◽  
Prime Divisors ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

This paper is really an attempt to solve the age-old problem of the Goldbach Conjecture, by restating it in terms of primes of the form 2x-q (where q is a prime less than or equal to x). Restating the problem merely requires us to ask the question: Does a prime of form 2x-q lie in the interval [x, 2x]? We begin by introducing the product, m, of numbers of the form 2x-q. Using the geometric series, an upper bound is estimated for the function m. Next, we prove a theorem that states every even number, 2x, that violates Goldbach’s Conjecture must satisfy an inequality involving a simple multiplicative function defined as the product, ρ(m), of the distinct prime divisors of m. A proof of the Goldbach Conjecture is then evident by contradiction as a corollary to the proof of the inequality.

Generating Functions for a Class of Arithmetic Functions

1966 ◽  
Vol 9 (4) ◽  
pp. 427-431 ◽  
Author(s):  
A. A. Gioia ◽  
M.V. Subbarao
Keyword(s):  
Generating Functions ◽  
Multiplicative Function ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Prime Divisors ◽  
Image Position

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity1.1

scholarly journals Distribution of the Largest Strong Goldbach Numbers Generated by Primes

2018 ◽  
Vol 10 (5) ◽  
pp. 1
Author(s):  
Pingyuan Zhou ◽  
Rong Ao
Keyword(s):  
Distribution Curve ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

Using the first 4000000 primes to find Ln, the largest strong Goldbach number generated by the n-th prime Pn, we generalize a proposition in our previous work (Zhou 2017) and propose that Ln ≈ 2Pn and Ln/2Pn < 1 for sufficiently large Pn but the limit of Ln/2Pn as n → ∞ is 1, Ln ≈ Pn + n log n and Ln/(Pn + n log n) > 1 for sufficiently large Pn but the limit of Ln/(Pn + n log n) as n → ∞ is 1. There are five corollaries of the generalized proposition for getting Ln → ∞ as n → ∞, which is equivalent to Goldbach’s conjecture. If every step in distribution curve of Ln is called a Goldbach step, a study on the ratio of width to height for Goldbach steps supports the existence of above two limits but a study on distribution of Goldbach steps supports an estimation that Q(n) ≈ (1 + 1/log log n)n/log n and the limit of Q(n)/((1 + 1/log log n)n/log n) as n → ∞ is 1, where Q(n) is the number of Goldbach steps, from which we may expect there are infinitely many Goldbach steps to imply Goldbach’s conjecture.

The Crystallographic Restriction, Permutations, and Goldbach's Conjecture

2003 ◽  
Vol 110 (3) ◽  
pp. 202 ◽  
Author(s):  
John Bamberg ◽  
Grant Cairns ◽  
Devin Kilminster
Keyword(s):  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

scholarly journals The Complexity of Mathematics

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Number Theory ◽  
Complexity Theory ◽  
Riemann Hypothesis ◽  
Open Problems ◽  
Mathematical Proofs ◽  
Pure Mathematics ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.

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