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Goldbach's conjecture is one of the oldest and best-known
unsolved problem List of unsolved problems may refer to several notable conjectures or open problems in various academic fields: Natural sciences, engineering and medicine * Unsolved problems in astronomy * Unsolved problems in biology * Unsolved problems in che ...
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: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...
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 × 1018, 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 N=p+1 larger than 4, for p 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 n, could not possibly rule out the existence of such a specific counterexample N). 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: Education * The Strong, an educational institution in Rochester, New York, United States * 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'' ≤ 105. 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 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 1/\ln m 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 1 \big/ \big ln m \, \ln(n - m)\big/math>. If one pursues this heuristic, one might expect the total number of ways to write a large even integer ''n'' as the sum of two odd primes to be roughly : \sum_^ \frac \frac \approx \frac. Since \ln n \ll \sqrt n, this quantity goes to infinity as ''n'' increases, and one would expect that every large even integer has not just one representation as the sum of two primes, but in fact very many such representations. This heuristic argument is actually somewhat inaccurate, because it assumes that the events of ''m'' and ''n'' − ''m'' being prime are statistically independent of each other. For instance, if ''m'' is odd, then ''n'' − ''m'' is also odd, and if ''m'' is even, then ''n'' − ''m'' is even, a non-trivial relation because, besides the number 2, only odd numbers can be prime. Similarly, if ''n'' is divisible by 3, and ''m'' was already a prime distinct from 3, then ''n'' − ''m'' would also be coprime to 3 and thus be slightly more likely to be prime than a general number. Pursuing this type of analysis more carefully, G. H. Hardy and John Edensor Littlewood in 1923 conjectured (as part of their '' Hardy–Littlewood prime tuple conjecture'') that for any fixed ''c'' ≥ 2, the number of representations of a large integer ''n'' as the sum of ''c'' primes n = p_1 + \cdots + p_c with p_1 \leq \cdots \leq p_c should be
asymptotically In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
equal to : \left(\prod_p \frac\right) \int_ \frac, where the product is over all primes ''p'', and \gamma_(n) is the number of solutions to the equation n = q_1 + \cdots + q_c \mod p in
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
, subject to the constraints q_1, \ldots, q_c \neq 0 \mod p. This formula has been rigorously proven to be asymptotically valid for ''c'' ≥ 3 from the work of Ivan Matveevich Vinogradov, but is still only a conjecture when c = 2. In the latter case, the above formula simplifies to 0 when ''n'' is odd, and to : 2 \Pi_2 \left(\prod_ \frac\right) \int_2^n \frac \approx 2 \Pi_2 \left(\prod_ \frac\right) \frac when ''n'' is even, where \Pi_2 is Hardy–Littlewood's twin prime constant : \Pi_2 := \prod_ \left(1 - \frac\right) \approx 0.66016 18158 46869 57392 78121 10014\dots This is sometimes known as the ''extended Goldbach conjecture''. The strong Goldbach conjecture is in fact very similar to the
twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin p ...
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 illustrate the above equations. See Goldbach's comet for more information. Goldbach's comet also suggests that there are tight upper and lower bounds on the number of representatives, and that the modulo ''6'' of ''2n'' plays a part in the number of representations. The number of representations is about n\ln n, from 2n = p + c and the Prime Number Theorem. If each ''c'' is composite, then it must have a prime factor less than or equal to the square root of 2n, by the method outlined in trial division. This leads to an expectation of \frac = \sqrt \frac\ln n representations.


Rigorous results

The strong Goldbach conjecture is much more difficult than the weak Goldbach conjecture. Using Vinogradov's method, Nikolai Chudakov,
Johannes van der Corput Johannes Gaultherus van der Corput (4 September 1890 – 16 September 1975) was a Dutch mathematician, working in the field of analytic number theory. He was appointed professor at the University of Fribourg (Switzerland) in 1922, at the Universi ...
, and
Theodor Estermann Theodor Estermann (5 February 1902 – 29 November 1991) was a German-born American mathematician, working in the field of analytic number theory. The Estermann measure, a measure of the central symmetry of a convex set in the Euclidean pl ...
showed that
almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathema ...
even numbers can be written as the sum of two primes (in the sense that the fraction of even numbers up to some N which can be so written tends towards 1 as N increases). In 1930,
Lev Schnirelmann Lev Genrikhovich Schnirelmann (also Shnirelman, Shnirel'man; ; 2 January 1905 – 24 September 1938) was a Soviet mathematician who worked on number theory, topology and differential geometry. Work Schnirelmann sought to prove Goldbach's conjec ...
proved that any
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 1 can be written as the sum of not more than prime numbers, where is an effectively computable constant; see Schnirelmann density. Schnirelmann's constant is the lowest number with this property. Schnirelmann himself obtained  < . This result was subsequently enhanced by many authors, such as Olivier Ramaré, who in 1995 showed that every even number 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, which directly implies that every even number is the sum of at most 4 primes. In 1924, Hardy and Littlewood showed under the assumption of the generalized Riemann hypothesis that the number of even numbers up to violating the Goldbach conjecture is much less than X^ for small . In 1948, using sieve theory,
Alfréd Rényi Alfréd Rényi (20 March 1921 – 1 February 1970) was a Hungarian mathematician known for his work in probability theory, though he also made contributions in combinatorics, graph theory, and number theory. Life Rényi was born in Budapest to A ...
showed that every sufficiently large even number can be written as the sum of a prime and an almost prime with at most K factors.
Chen Jingrun Chen Jingrun (; 22 May 1933 – 19 March 1996), also known as Jing-Run Chen, was a Chinese mathematician who made significant contributions to number theory, including Chen's theorem and the Chen prime. Life and career Chen was the third son i ...
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 In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime ...
(the product of two primes). See
Chen's theorem In number theory, Chen's theorem states 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). History The theorem was first stated by Chinese mathema ...
for further information. In 1975, Hugh Montgomery and Robert Charles Vaughan showed that "most" even numbers are expressible as the sum of two primes. More precisely, they showed that there exist positive constants and such that for all sufficiently large numbers , every even number less than is the sum of two primes, with at most C N^ exceptions. In particular, the set of even integers that are not the sum of two primes has
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
zero. In 1951, Yuri Linnik proved the existence of a constant such that every sufficiently large even number is the sum of two primes and at most powers of 2.
Roger Heath-Brown David Rodney "Roger" Heath-Brown FRS (born 12 October 1952), is a British mathematician working in the field of analytic number theory. Education He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervi ...
and Jan-Christoph Schlage-Puchta found in 2002 that works.


Related problems

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. Similar problems to Goldbach's conjecture exist in which primes are replaced by other particular sets of numbers, such as the squares: * It was proven by Lagrange that every positive integer is the sum of four squares. See Waring's problem and the related Waring–Goldbach problem on sums of powers of primes. * Hardy and Littlewood listed as their Conjecture I: "Every large odd number (''n'' > 5) is the sum of a prime and the double of a prime" ('' Mathematics Magazine'', 66.1 (1993): 45–47). This conjecture is known as Lemoine's conjecture and is also called ''Levy's conjecture''. * The Goldbach conjecture for practical numbers, a prime-like sequence of integers, was stated by Margenstern in 1984, and proved by Melfi in 1996: every even number is a sum of two practical numbers. * A strengthening of the Goldbach conjecture proposed by Harvey Dubner states that every even integer greater than 4,208 is the sum of two
twin prime A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin p ...
s. Only 34 even integers less than 4,208 are not the sum of two twin primes. Dubner has verified computationally that this list is complete up to . A proof of this stronger conjecture would not only imply Goldbach's conjecture, but also the twin prime conjecture.


In popular culture

''Goldbach's Conjecture'' () is the title of the biography of Chinese mathematician and number theorist
Chen Jingrun Chen Jingrun (; 22 May 1933 – 19 March 1996), also known as Jing-Run Chen, was a Chinese mathematician who made significant contributions to number theory, including Chen's theorem and the Chen prime. Life and career Chen was the third son i ...
, written by Xu Chi. The conjecture is a central point in the plot of the 1992 novel '' Uncle Petros and Goldbach's Conjecture'' by Greek author Apostolos Doxiadis, in the short story " Sixty Million Trillion Combinations" by
Isaac Asimov yi, יצחק אזימאװ , birth_date = , birth_place = Petrovichi, Russian SFSR , spouse = , relatives = , children = 2 , death_date = , death_place = Manhattan, New York City, U.S. , nationality = Russian (1920–1922)Soviet (192 ...
and also in the 2008 mystery novel ''No One You Know'' by Michelle Richmond. Goldbach's conjecture is part of the plot of the 2007 Spanish film '' Fermat's Room''.


References


Further reading

* *
Terence Tao proved that all odd numbers are at most the sum of five primes


at MathWorld.


External links

* *
Goldbach's original letter to Euler — PDF format (in German and Latin)''Goldbach's conjecture''
part of Chris Caldwell's Prime Pages.
''Goldbach conjecture verification''
Tomás Oliveira e Silva's distributed computer search. {{DEFAULTSORT:Goldbach's Conjecture Additive number theory Analytic number theory Conjectures about prime numbers Unsolved problems in number theory Hilbert's problems