Goldbach's conjecture is one of the oldest and best-known
unsolved problem
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s in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
and all of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. It states that every
even
Even may refer to:
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* Even (surname)
* Even (people), an ethnic group from Siberia and Russian Far East
**Even language, a language spoken by the Evens
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natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
greater than 2 is the sum of two
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s.
The conjecture has been shown to hold for all integers less than 4 × 10
18, but remains unproven despite considerable effort.
History
On 7 June 1742, the German mathematician
Christian Goldbach
Christian Goldbach (; ; 18 March 1690 – 20 November 1764) was a German mathematician connected with some important research mainly in number theory; he also studied law and took an interest in and a role in the Russian court. After traveling ...
wrote a letter to
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
(letter XLIII), in which he proposed the following conjecture:
Goldbach was following the now-abandoned convention of
considering 1 to be a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
,
so that a sum of units would indeed be a sum of primes.
He then proposed a second conjecture in the margin of his letter, which implies the first:
Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had had (), in which Goldbach had remarked that the first of those two conjectures would follow from the statement
This is in fact equivalent to his second, marginal conjecture.
In the letter dated 30 June 1742, Euler stated:
Each of the three conjectures above has a natural analog in terms of the modern definition of a prime, under which 1 is excluded.
A modern version of the first conjecture is:
A modern version of the marginal conjecture is:
And a modern version of Goldbach's older conjecture of which Euler reminded him is:
These modern versions might not be entirely equivalent to the corresponding original statements. For example, if there were an even integer
larger than 4, for
a prime, that could not be expressed as the sum of two primes in the modern sense, then it would be a counterexample to the modern version of the third conjecture (without being a counterexample to the original version). The modern version is thus probably stronger (but in order to confirm that, one would have to prove that the first version, freely applied to any positive even integer
, could not possibly rule out the existence of such a specific counterexample
). In any case, the modern statements have the same relationships with each other as the older statements did. That is, the second and third modern statements are equivalent, and either implies the first modern statement.
The third modern statement (equivalent to the second) is the form in which the conjecture is usually expressed today. It is also known as the "
strong
Strong may refer to:
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* Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas
* Strong School, New Haven, Connecticut, United S ...
", "even", or "binary" Goldbach conjecture. A weaker form of the second modern statement, known as "
Goldbach's weak conjecture", the "odd Goldbach conjecture", or the "ternary Goldbach conjecture", asserts that
A proof for the weak conjecture was proposed in 2013 by
Harald Helfgott. Helfgott's proof has not yet appeared in a peer-reviewed publication, though was accepted for publication in the ''
Annals of Mathematics Studies
Annals ( la, annāles, from , "year") are a concise historical record in which events are arranged chronologically, year by year, although the term is also used loosely for any historical record.
Scope
The nature of the distinction between anna ...
'' series in 2015 and has been undergoing further review and revision since.
The weak conjecture would be a corollary of the strong conjecture: if is a sum of two primes, then is a sum of three primes. However, the converse implication and thus the strong Goldbach conjecture remain unproven.
Verified results
For small values of ''n'', the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, in 1938, Nils Pipping laboriously verified the conjecture up to ''n'' ≤ 10
5. 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 × 10
18 (and double-checked up to 4 × 10
17) as of 2013. One record from this search is that is the smallest number that cannot be written as a sum of two primes where one is smaller than 9781.
Heuristic justification
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.
A very crude version of the
heuristic
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
probabilistic argument (for the strong form of the Goldbach conjecture) is as follows. The
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying t ...
asserts that an integer ''m'' selected at random has roughly a
chance of being prime. Thus if ''n'' is a large even integer and ''m'' is a number between 3 and ''n''/2, then one might expect the probability of ''m'' and ''n'' − ''m'' simultaneously being prime to be