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Title: On the Goldbach Conjecture

Abstract

This note proposes a confirmation proof of Goldbach’s Conjecture from 1742 [gol]: Can every even integer n > 2 be represented as the sum of 2 prime numbers? First, I present an analysis of the nature of multiplication as description length reduction for addition. This allows to explain prime numbers from first principles rather than by definition. Then, I use this analysis to derive that any natural number n must be representable by n – 1 unique two-summand additions. Given that an explicit formula for the number of primes smaller n is unknown, I formulate the contraposition to the conjecture. I then count the maximum number of possible unique additions when the summands are both non-prime or 1 and when one summand is non-prime, prime or 1 and the other is non-prime or 1. This is done using a case distinction for even and odd n. The counting is facilitated by finding the maximum number of positions a plus symbol can be placed into the unary representation of n, given the constraints. The main result is that no even n ≥ 8 exists where all n – 1 additions can be formulated without including the case of the addition ofmore » two primes. A side result is that any odd integer n can be represented by 1 addition of 2 primes if and only if n – 2 is prime.« less

Authors:
 [1]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1476233
Report Number(s):
LLNL-TR-748717
931561
DOE Contract Number:  
AC52-07NA27344
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING

Citation Formats

Friedland, Gerald. On the Goldbach Conjecture. United States: N. p., 2018. Web. doi:10.2172/1476233.
Friedland, Gerald. On the Goldbach Conjecture. United States. doi:10.2172/1476233.
Friedland, Gerald. Thu . "On the Goldbach Conjecture". United States. doi:10.2172/1476233. https://www.osti.gov/servlets/purl/1476233.
@article{osti_1476233,
title = {On the Goldbach Conjecture},
author = {Friedland, Gerald},
abstractNote = {This note proposes a confirmation proof of Goldbach’s Conjecture from 1742 [gol]: Can every even integer n > 2 be represented as the sum of 2 prime numbers? First, I present an analysis of the nature of multiplication as description length reduction for addition. This allows to explain prime numbers from first principles rather than by definition. Then, I use this analysis to derive that any natural number n must be representable by n – 1 unique two-summand additions. Given that an explicit formula for the number of primes smaller n is unknown, I formulate the contraposition to the conjecture. I then count the maximum number of possible unique additions when the summands are both non-prime or 1 and when one summand is non-prime, prime or 1 and the other is non-prime or 1. This is done using a case distinction for even and odd n. The counting is facilitated by finding the maximum number of positions a plus symbol can be placed into the unary representation of n, given the constraints. The main result is that no even n ≥ 8 exists where all n – 1 additions can be formulated without including the case of the addition of two primes. A side result is that any odd integer n can be represented by 1 addition of 2 primes if and only if n – 2 is prime.},
doi = {10.2172/1476233},
journal = {},
number = ,
volume = ,
place = {United States},
year = {2018},
month = {2}
}