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Goldbach problem

In mathematics, 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.)

For example,

  4 = 2 + 2

  6 = 3 + 3

  8 = 3 + 5

10 = 3 + 7 = 5 + 5

12 = 5 + 7

14 = 3 + 11 = 7 + 7

etc.

Origins

In 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he proposed the following conjecture:

Every number greater than 5 can be written as the sum of three primes.

Euler, becoming interested in the problem, answered with a stronger version of the conjecture:

Every even number greater than 2 can be written as the sum of two primes.

The former conjecture is known today as the 'weak' Goldbach conjecture, the latter as the 'strong' Goldbach conjecture. (The strong version implies the weak version, as any odd number greater than 5 can be obtained by adding 3 to any even number greater than 2). Without qualification, the strong version is meant. Both questions have remained unsolved ever since.

Results

Goldbach's conjecture has been researched by many number theorists. The majority of mathematicians believe the (strong) 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.

Some progress was made in the 1930s. First, in 1937, Ivan Vinogradov proved that every odd number <math>n>3^{3^{15}}<math> is the sum of three primes, and 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 1938, Theodor Estermann (1902-1991) proved that almost all even numbers are the sum of two primes, and N. Pipping laboriously verified the conjecture for all n ≤ 10,000. L.G. Schnirelmann subsequently proved in 1939 that every even number n ≥ 4 can be written as the sum of at most 300,000 primes (later lowered to the sum of at most 7 primes).

Later mathematicians have developed other approaches. One method endeavors to prove that "every even number greater than 4 can be written as the sum of c primes". A further generalization along these lines would be to prove that "every multiple of c greater than c itself can be written as the sum of c primes". According to either of these formulations, Goldbach's conjecture is the special case where c=2. Another method attempts to prove that "every even number can be written as the sum of a number whose prime factors are no more than a and a number whose prime factors are no more than b". This is called the "(a+b) proposition". In this approach, Goldbach's conjecture is the special case where a=1 and b=1; that is, when both numbers are prime. Goldbach's would therefore be the "(1+1) proposition".

Chen Jingrun showed in 1966 that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes)—e.g., 100 = 23 + 7·11.

H.A. Pogorzelski circulated a proof of the Goldbach conjecture in 1977, but this work is not generally accepted in mathematical circles.

T. Oliveira e Silva is running a distributed computer search that has verified the conjecture up to 2 × 10¹⁷ (as of March 2004).

Viktar Karpau (Victor Karpov), a mathematician from Belarus, allegedly found a proof of Goldbach's conjecture which was published in September 2004.

Trivia

Doug Lenat's Automated Mathematician rediscovered Goldbach's Conjecture in 1982. This is considered one of the earliest demonstrations that artificial intelligences are capable of scientific discovery (but see the discussion at Automated Mathematician).

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.

A student at the University Of London has claimed that he has found proof of the Conjecture. Andrew Wiles (editor of Annals of Mathematics) has seen part of the proof, and has said it looks very promising.