The Goldbach conjecture, also called the "strong" or "binary" Goldbach conjecture, asserts that every positiveeveninteger can be expressed as the sum
of two primes. A pair of primes satisfying is called a Goldbach
partition of
(Oliveira e Silva). This conjecture remains open.
The Goldbach partitions of the even numbers through 30 are illustrated above. The red and blue lines fix the two prime summands,
and each green diamond at an intersection is labeled by their sum (Veritasium 2025).
Goldbach's original formulation, written in a June 7, 1742 letter to Euler, stated "at least it seems that every number that is greater than 2 is the sum
of three primes" (Goldbach 1742; Dickson 2005,
p. 421). Goldbach counted 1 as a prime, a convention
no longer followed. Under that convention, Euler's binary restatement was equivalent
to Goldbach's original assertion. The related weak
Goldbach conjecture, concerning sums of three primes, is now a theorem.
Chen Jingrun's schoolteacher Shen Yuan memorably described mathematics as the queen of science, number theory as its crown,
and Goldbach's conjecture as the pearl in the crown. Chen later recalled that he
never forgot this characterization (O'Connor and Robertson 2022).
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 twoprimes!
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 .
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 oddprimes,
and every integer the sum of exactly three distinct primes.
can be expressed as the sum
of two primes 
. A pair of primes 
satisfying 
is called a Goldbach
partition of 
(Oliveira e Silva). This conjecture remains open.
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.
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 
.
,
there are primes 
and 
such that
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).
is the sum of two odd primes,
and every integer 
the sum of exactly three distinct primes.
." Oct. 3, 2003b. 
." Math. Comput. 70, 1745-1750,
2001.Schnirelman, L. G. "On the Additive Properties of Numbers."
[Russian]. Uspekhi Mat. Nauk No. 6, 9-25, 1939. 
."
Math. Comput. 61, 931-934, 1993.Stein, M. L. and
Stein, P. R. "New Experimental Results on the Goldbach Conjecture."
Math. Mag. 38, 72-80, 1965a.Stein, M. L. and Stein,
P. R. "Experimental Results on Additive 2 Bases." BIT 38,
427-434, 1965b.Veritasium. "The Obviously True Theorem No One Can
Prove." Jun. 20, 2025. 