Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states:

The conjecture has been shown to hold for all integers less than 4 × 1018,^[2] but remains unproven despite considerable effort.

A Goldbach number is a positive even integer that can be expressed as the sum of two odd primes.^[4] Since 4 is the only even number greater than 2 that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers.

The expression of a given even number as a sum of two primes is called a Goldbach partition of that number. The following are examples of Goldbach partitions for some even numbers:

The number of ways in which 2n can be written as the sum of two primes (for n starting at 1) is:

On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII),^[6] in which he proposed the following conjecture:

He then proposed a second conjecture in the margin of his letter:

He considered 1 to be a prime number, a convention subsequently abandoned.^[1] The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time. A modern version of Goldbach's marginal conjecture is:

Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had ("…so Ew vormals mit mir communicirt haben…"), in which Goldbach remarked his original (and not marginal) conjecture followed from the following statement

which is, thus, also a conjecture of Goldbach. In the letter dated 30 June 1742, Euler stated:^[7][8]

Goldbach's third version (equivalent to the two other versions) is the form in which the conjecture is usually expressed today. It is also known as the "strong", "even", or "binary" Goldbach conjecture, to distinguish it from a weaker conjecture, known today variously as the Goldbach's weak conjecture, the "odd" Goldbach conjecture, or the "ternary" Goldbach conjecture. This weak conjecture asserts that all odd numbers greater than 7 are the sum of three odd primes and appears to have been proved in 2013.^[9][10] The weak conjecture is a corollary of the strong conjecture: if n – 3 is a sum of two primes, then n is a sum of three primes. The converse implication and the strong Goldbach conjecture remain unproven.

For small values of n, the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, Nils Pipping in 1938 laboriously verified the conjecture up to n ≤ 105.^[11] With the advent of computers, many more values of n have been checked; T. Oliveira e Silva ran a distributed computer search that has verified the conjecture for n ≤ 4 × 1018 (and double-checked up to 4 × 1017) as of 2013. One record from this search is that 3325581707333960528 is the smallest number that has no Goldbach partition with a prime below 9781.^[12]

Statistical considerations that focus on the probabilistic distribution of prime numbers present informal evidence in favour of the conjecture (in both the weak and strong forms) for sufficiently large integers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, and the more "likely" it becomes that at least one of these representations consists entirely of primes.

Since this quantity goes to infinity as n increases, we expect that every large even integer has not just one representation as the sum of two primes, but in fact has very many such representations.

This is sometimes known as the extended Goldbach conjecture. The strong Goldbach conjecture is in fact very similar to the twin prime conjecture, and the two conjectures are believed to be of roughly comparable difficulty.

The Goldbach partition functions shown here can be displayed as histograms, which informatively illustrate the above equations. See Goldbach's comet.^[13]

The strong Goldbach conjecture is much more difficult than the weak Goldbach conjecture. Using Vinogradov's method, Chudakov,^[14] Van der Corput,^[15] and Estermann^[16] showed 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 1930, Lev Schnirelmann proved^[17][18] that any natural number greater than 1 can be written as the sum of not more than C prime numbers, where C is an effectively computable constant, see Schnirelmann density. Schnirelmann's constant is the lowest number C with this property. Schnirelmann himself obtained C < 800000. This result was subsequently enhanced by many authors, such as Olivier Ramaré, who in 1995 showed that every even number n ≥ 4 is in fact the sum of at most 6 primes. The best known result currently stems from the proof of the weak Goldbach conjecture by Harald Helfgott,^[19] which directly implies that every even number n ≥ 4 is the sum of at most 4 primes.^[20][21]

Chen Jingrun showed in 1973 using the methods of sieve theory 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).^[23] See Chen's theorem for further information.

In 1951, Linnik proved the existence of a constant K such that every sufficiently large even number is the sum of two primes and at most K powers of 2. Roger Heath-Brown and Jan-Christoph Schlage-Puchta in 2002 found that K = 13 works.^[24]

As with many famous conjectures in mathematics, there are a number of purported proofs of the Goldbach conjecture, none of which are accepted by the mathematical community.

Although Goldbach's conjecture implies that every positive integer greater than one can be written as a sum of at most three primes, it is not always possible to find such a sum using a greedy algorithm that uses the largest possible prime at each step. The Pillai sequence tracks the numbers requiring the largest number of primes in their greedy representations.^[25]

One can consider similar problems in which primes are replaced by other particular sets of numbers, such as the squares.

The conjecture is a central point in the plot of the 1992 novel Uncle Petros and Goldbach's Conjecture by Greek author Apostolos Doxiadis.