# 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 »

- Authors:

- 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}

}