Goldbach's original conjecture (sometimes called the "ternary" Goldbach conjecture), written in a June 7, 1742 letter to Euler, states "at least it seems that every number that is greater than 2 is the sum of three primes" (Goldbach 1742; Dickson 2005, p. 421). Note that Goldbach considered the number 1 to be a prime, a convention that is no longer followed. As re-expressed by Euler, an equivalent form of this conjecture (called the "strong" or "binary" Goldbach conjecture) asserts that all positive even integers can be expressed as the sum of two primes. Two primes such that for a positive integer are sometimes called a Goldbach partition (Oliveira e Silva).
According to Hardy (1999, p. 19), "It is comparatively easy to make clever guesses; indeed there are theorems, like 'Goldbach's Theorem,' which have never been proved and which any fool could have guessed." Faber and Faber offered a prize to anyone who proved Goldbach's conjecture between March 20, 2000 and March 20, 2002, but the prize went unclaimed and the conjecture remains open.
Schnirelman (1939) proved that every even number can be written as the sum of not more than primes (Dunham 1990), which seems a rather far cry from a proof for two primes! Pogorzelski (1977) claimed to have proven the Goldbach conjecture, but his proof is not generally accepted (Shanks 1985). The following table summarizes bounds such that the strong Goldbach conjecture has been shown to be true for numbers .
|Stein and Stein 1965ab|
|Granville et al. 1989|
|Deshouillers et al. 1998|
|Richstein 1999, 2001|
|Oliveira e Silva (Mar. 24, 2003)|
|Oliveira e Silva (Oct. 3, 2003)|
|Oliveira e Silva (Feb. 5, 2005)|
|Oliveira e Silva (Dec. 30, 2005)|
|Oliveira e Silva (Jul. 14, 2008)|
|Oliveira e Silva (Apr. 2012)|
The conjecture that all odd numbers are the sum of three odd primes is called the "weak" Goldbach conjecture. Vinogradov (1937ab, 1954) proved that every sufficiently large odd number is the sum of three primes (Nagell 1951, p. 66; Guy 1994), and Estermann (1938) proved that almost all even numbers are the sums of two primes. Vinogradov's original "sufficiently large" was subsequently reduced to by Chen and Wang (1989). Chen (1973, 1978) also showed that all sufficiently large even numbers are the sum of a prime and the product of at most two primes (Guy 1994, Courant and Robbins 1996). More than two and a half centuries after the original conjecture was stated, the weak Goldbach conjecture was proved by Helfgott (2013, 2014).
A stronger version of the weak conjecture, namely that every odd number can be expressed as the sum of a prime plus twice a prime is known as Levy's conjecture.
An equivalent statement of the Goldbach conjecture is that for every positive integer , there are primes and such that
where is the totient function (e.g., Havil 2003, p. 115; Guy 2004, p. 160). (This follows immediately from for prime.) Erdős and Moser have considered dropping the restriction that and be prime in this equation as a possibly easier way of determining if such numbers always exist (Guy 1994, p. 105).
Other variants of the Goldbach conjecture include the statements that every even number is the sum of two odd primes, and every integer the sum of exactly three distinct primes.
Let be the number of representations of an even number as the sum of two primes. Then the "extended" Goldbach conjecture states that
where is the twin primes constant (Halberstam and Richert 1974).