Goldbach's Conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:
(The same prime may be used twice.) The conjecture had been known to Descartes. The following statement is equivalent and is the one originally conjectured in a letter written by Goldbach to Euler in 1742:
We know that every even number can be written as the sum of at most six primes. As a result of work by Vinogradov, every sufficiently large even number can be written as the sum of at most four primes. Vinogradov proved furthermore that almost all even numbers can be written as the sum of two primes (in the sense that the fraction of even numbers which can be so written tends towards 1). In 1966, Chen Jing-run showed that every sufficiently large even number can be written as the sum of a prime and a number with at most two prime factors.
In 1982 Doug Lenat's Automated Mathematician independently rediscovered Goldbach's Conjecture in one of the earliest demonstrations that Artificial Intelligences were capable of scientific discovery.
In order to generate publicity for the book Uncle Petros and Goldbach's Conjecture by Apostolos Doxiadis, British publisher Tony Faber offered a $1,000,000 prize for a proof of the conjecture in 2000. The prize was only to be paid for proofs submitted for publication before April 2002. The prize was never claimed.
Goldbach made two related conjectures about sums of primes, the 'strong' Goldbach conjecture and the 'weak' Goldbach conjecture. The conjecture merely referred to as "Goldbach's conjecture" is the strong one which is discussed here.
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