Goldbach’s Conjecture: A Million Dollar Question

The main mathematical news

Late 20th century: Prime numbers are used to encrypt information over the Internet (e.g., online shopping).

2003: Every even number is the sum of two primes and at most eight powers of 2.

2013: (using computers) –  every even number greater than 2 and smaller than  can be expressed as the sum of two primes

2013: Proof of Goldbach’s weak conjecture – Every odd number greater than 5 can be expressed as the sum of three primes.

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Additional Theorems / conjectures / Open questions

* There are infinitely many primes

* Goldbach’s conjecture: Every even number (greater than 2) can be expressed as the sum of two primes.

* Support: Every known even number can be expressed as a sum of two primes.

* The fraction of even numbers smaller than N which cannot be expressed this way tends towards 0 when N→∞.

* Any even number can be written as the sum of not more than 6 primes.

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The main mathematical concepts / Principles

Number theory (MSC2010#97F60)

*    Prime/Composite (natural) number

*    Semiprime

Logic (MSC2010#97E30)

*        Theorem

*        Conjecture

*        Proof

*        Refutation

*        Open problem

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