Goldbach's Conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:
Every even number greater than 2 can be written as the sum of two primes.
(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:
Every number greater than 5 can be written as the sum of three primes.
This conjecture has been researched by many number theorists and has been checked by computer for even numbers up to 4 × 10^{14}. The majority of mathematicians believe the conjecture to be true, mostly based on statistical considerations focusing on the probabilistic distribution of prime numbers: the bigger the even number, the more "likely" it becomes that it can be written as a sum of two primes.
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 Jingrun[?] 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 independentally 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 Doxiadisone, 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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