Goldbach Conjecture
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Goldbach's conjecture is one of the oldest and best-known unsolved problems 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 arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
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 w ...
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 n ...
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 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 because the only ways ...
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 in ma ...
(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 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 because the only ways ...
, 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", "even", or "binary" Goldbach conjecture. A weaker form of the second modern statement, known as "
Goldbach's weak conjecture In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that : Every odd number greater than 5 can be expressed as the sum of three primes. (A prime ma ...
", the "odd Goldbach conjecture", or the "ternary Goldbach conjecture", asserts that A proof for the weak conjecture was proposed in 2013 by
Harald Helfgott Harald Andrés Helfgott (born 25 November 1977) is a Peruvian mathematician working in number theory. Helfgott is a researcher ('' directeur de recherche'') at the CNRS at the Institut Mathématique de Jussieu, Paris. Early life and education ...
. Helfgott's proof has not yet appeared in a peer-reviewed publication, though was accepted for publication in the '' Annals of Mathematics Studies'' 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 In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have pa ...
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 ...
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 Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...
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 In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
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 Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
and
John Edensor Littlewood John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations, and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...
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 book ...
, 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 Ivan Matveevich Vinogradov ( rus, Ива́н Матве́евич Виногра́дов, p=ɪˈvan mɐtˈvʲejɪvʲɪtɕ vʲɪnɐˈɡradəf, a=Ru-Ivan_Matveyevich_Vinogradov.ogg; 14 September 1891 – 20 March 1983) was a Soviet mathematician, ...
, 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 pr ...
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 Goldbach's comet is the name given to a plot of the function g(E), the so-called Goldbach function . The function, studied in relation to Goldbach's conjecture, is defined for all even integers E > 2 to be the number of different ways in which ''E ...
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 Trial division is the most laborious but easiest to understand of the integer factorization algorithms. The essential idea behind trial division tests to see if an integer ''n'', the integer to be factored, can be divided by each number in turn ...
. 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 In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that : Every odd number greater than 5 can be expressed as the sum of three primes. (A prime m ...
. Using Vinogradov's method,
Nikolai Chudakov Nikolai Grigor'evich Chudakov (russian: Никола́й Григо́рьевич Чудако́в; 1904–1986) was a Russian and Soviet mathematician. He was born in Lysovsk, Novo-Burassk, Saratov, Russian Empire. His father worked as a medical a ...
,
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 Univers ...
, 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 plan ...
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 math ...
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 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 n ...
greater than 1 can be written as the sum of not more than prime numbers, where is an effectively computable constant; see
Schnirelmann density In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it.Schnirelmann, L.G. (1930).On the ...
. 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 Olivier Ramaré is a French mathematician who works as Senior researcher for the CNRS. He is currently attached to Aix-Marseille Université. Ramaré earned a doctorate in 1991 from the University of Bordeaux with a dissertation ''Contribution au ...
, 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 Harald Andrés Helfgott (born 25 November 1977) is a Peruvian mathematician working in number theory. Helfgott is a researcher ('' directeur de recherche'') at the CNRS at the Institut Mathématique de Jussieu, Paris. Early life and education ...
, 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 The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whic ...
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 ...
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 in ...
showed in 1973 using the methods of
sieve theory Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed lim ...
that every
sufficiently large In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have pa ...
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 nu ...
(the product of two primes). See
Chen's theorem In number theory, Chen's theorem states that every sufficiently large parity (mathematics), even number can be written as the sum of either two prime number, primes, or a prime and a semiprime (the product of two primes). History The theorem wa ...
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. Mathematical ...
zero. In 1951,
Yuri Linnik Yuri Vladimirovich Linnik (russian: Ю́рий Влади́мирович Ли́нник; January 8, 1915 – June 30, 1972) was a Soviet mathematician active in number theory, probability theory and mathematical statistics. Linnik was born in B ...
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 supervis ...
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 A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally ...
that uses the largest possible prime at each step. The
Pillai sequence The Pillai sequence is the sequence of integers that have a record number of terms in their greedy representations as sums of prime numbers (and one). It is named after Subbayya Sivasankaranarayana Pillai, who first defined it in 1930. It would fo ...
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 In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural num ...
and the related
Waring–Goldbach problem The Waring–Goldbach problem is a problem in additive number theory, concerning the representation of integers as sums of powers of prime numbers. It is named as a combination of Waring's problem on sums of powers of integers, and the Goldbach con ...
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 ''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a j ...
'', 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 number In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. For example, 12 is a practical number because all the numbers from 1 ...
s, a prime-like sequence of integers, was stated by Margenstern in 1984, and proved by
Melfi Melfi (Neapolitan language, Lucano: ) is a town and ''comune'' in the Vulture area of the province of Potenza, in the Southern Italian region of Basilicata. Geographically, it is midway between Naples and Bari. In 2015 it had a population of 17,7 ...
in 1996: every even number is a sum of two practical numbers. * A strengthening of the Goldbach conjecture proposed by
Harvey Dubner Harvey Dubner (1928–2019) was an electrical engineer and mathematician who lived in New Jersey, noted for his contributions to finding large prime numbers. In 1984, he and his son Robert collaborated in developing the 'Dubner cruncher', a board ...
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 pr ...
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 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 pr ...
.


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 in ...
, 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 Apostolos K. Doxiadis ( el, Απόστολος Κ. Δοξιάδης; born 1953) is a Greek writer. He is best known for his international bestsellers '' Uncle Petros and Goldbach's Conjecture'' (2000) and ''Logicomix'' (2009). Early life Doxiad ...
, in the short story "
Sixty Million Trillion Combinations "Sixty Million Trillion Combinations" is a short mystery story by American writer Isaac Asimov. It was first published in the May 5, 1980, issue of ''Ellery Queen's Mystery Magazine'' under the title "64 Million Trillion Combinations," and reprinted ...
" 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 Michelle Richmond is an American novelist, essayist, and short story writer. She wrote ''The Year of Fog'', which was a New York Times bestseller,''The Marriage Pact'', which was a Sunday Times bestseller, and six other books of fiction. Biograph ...
. Goldbach's conjecture is part of the plot of the 2007 Spanish film ''
Fermat's Room ''Fermat's Room'' ( es, La habitación de Fermat) is a 2007 Spanish thriller film directed by Luis Piedrahita and Rodrigo Sopeña. Three mathematicians and one inventor are invited to a house under the premise of solving a great enigma, and told ...
''.


References


Further reading

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


at
MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...
.


External links

* *
Goldbach's original letter to Euler — PDF format (in German and Latin)''Goldbach's conjecture''
part of Chris Caldwell's
Prime Pages The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" ...
.
''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