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 here 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  .
| bound | reference |  | Desboves 1885 |  | Pipping 1938 |  | Stein and Stein 1965ab |  | Granville et
al. 1989 |  | Sinisalo 1993 |  | 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) |
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).
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).
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157-176, 1973.
Chen, J. R. "On the Representation of a Large Even Integer as the Sum of a Prime and the Product of at Most Two Primes, II." Sci. Sinica 21,
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Study Institute held in Banff, Alberta, April 27-May 5, 1988 (Ed. R. A. Mollin).
Dordrecht, Netherlands: Kluwer, pp. 423-433, 1989.
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pp. 105-107, 1994.
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2004.
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Nagell, T. Introduction to Number Theory. New York: Wiley, p. 66,
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Oliveira e Silva, T. "Verification of the Goldbach Conjecture Up to  ."
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