Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states: Unsolved problems in : Note: Use the unsolved tag: {{unsolved|F|X}}, where F is any field in the sciences: and X is a concise explanation with or without links. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... For other meanings of mathematics or math, see mathematics (disambiguation). ...

Every even integer greater than 2 can be written as the sum of two primes.

For example, In mathematics, the parity of an object refers to whether it is even or odd. ... The integers are commonly denoted by the above symbol. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. ...

  4 = 2 + 2

  6 = 3 + 3

  8 = 3 + 5

10 = 3 + 7 = 5 + 5

12 = 5 + 7

14 = 3 + 11 = 7 + 7

etc.

Origins

On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) [1] in which he proposed the following conjecture: June 7 is the 158th day of the year in the Gregorian calendar (159th in leap years), with 207 days remaining. ... // Events January 24 - Charles VII Albert becomes Holy Roman Emperor. ... Flag of Prussia (1894 - 1918) The Kingdom of Prussia existed from 1701 until 1918, and from 1871 was the leading kingdom of the German Empire, comprising in its last form almost two-thirds of the area of the Empire. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Christian Goldbach (March 18, 1690 - November 20, 1764), was a Prussian mathematician, who was born in Königsberg, Prussia, as son of a pastor. ... Euler redirects here. ...

Every integer greater than 2 can be written as the sum of three primes.

He considered 1 to be a prime number, a convention subsequently abandoned. So today, Goldbach's original conjecture would be written: In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors. ...

Every integer greater than 5 can be written as the sum of three primes.

Euler, becoming interested in the problem, answered with an equivalent version of the conjecture:

Every even integer greater than 2 can be written as the sum of two primes,

adding that he regarded this a fully certain theorem ("ein ganz gewisses Theorema"), in spite of his being unable to prove it.

The former conjecture is today known as the "ternary" Goldbach conjecture, the latter as the "strong" or "binary" Goldbach conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is called the "weak" Goldbach conjecture. Both questions have remained unsolved ever since, although the weak form of the conjecture is much closer to resolution than the strong one. In number theory, Goldbachs weak conjecture, also known as the odd Goldbach conjecture or the 3-primes problem, states that: Every odd number greater than 7 can be expressed as the sum of three odd primes. ...

Heuristic justification

The majority of mathematicians believe the conjecture (in both the weak and strong forms) to be true, at least for sufficiently large integers, mostly based on statistical considerations focusing on the probabilistic distribution of prime numbers: the bigger the number, 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. In mathematics, the phrase sufficiently large is used in contexts such as: is true for sufficiently large which is actually shorthand for: there exists an such that is true for all . ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...

Number of ways to write an even number n as the sum of two primes (4 ≤ n ≤ 1,000)

Number of ways to write an even number n as the sum of two primes (4 ≤ n ≤ 1,000,000)

A very crude version of the heuristic probabilistic argument (for the strong form of the Goldbach conjecture) is as follows. The prime number theorem 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 . This heuristic is non-rigorous for a number of reasons, for instance it assumes that the events that m and n − m are prime are statistically independent of each other. Nevertheless, 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 Image File history File links Goldbach-1000. ... Image File history File links Goldbach-1000. ... Image File history File links Goldbach-1000000. ... Image File history File links Goldbach-1000000. ... Heuristic is the art and science of discovery and invention. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ... In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs. ...

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.

The above heuristic argument is actually somewhat inaccurate, because it ignores some correlations between the likelihood of m and n − m being prime. 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 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, Hardy and Littlewood in 1923 conjectured (as part of their famous 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 with should be asymptotically equal to Linear correlations between 1000 pairs of numbers. ... Coprime - Wikipedia /**/ @import /skins-1. ... G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ... John Edensor Littlewood (June 9, 1885 – September 6, 1977) was a British mathematician. ... 1923 (MCMXXIII) was a common year starting on Monday (link will take you to calendar). ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ...

where the product is over all primes p, and γ_c,p(n) is the number of solutions to the equation in modular arithmetic, subject to the constraints . This formula has been rigorously proven to be asymptotically valid for c ≥  3 from the work of 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 Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... A constraint is a limitation of possibilities. ... The neutrality of this article is disputed. ...

when n is even, where Π₂ is the twin prime constant The twin prime conjecture is a famous problem in number theory that involves prime numbers. ...

This asymptotic 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. In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ... The twin prime conjecture is a famous problem in number theory that involves prime numbers. ...

Rigorous results

For small values of n, the strong Goldbach conjecture (and hence the weak Goldbach conjecture) can be verified directly. For instance, N. Pipping in 1938 laboriously verified the conjecture up to . With the advent of computers, many more small values of n have been checked; T. Oliveira e Silva is running a distributed computer search that has verified the conjecture up to (as of June 2006). Year 1938 (MCMXXXVIII) was a common year starting on Saturday (link will take you to calendar). ... 2006 is a common year starting on Sunday of the Gregorian calendar. ...

The weak Goldbach conjecture is fairly close to resolution. In 1923, Hardy and Littlewood showed that under the assumption of the generalized Riemann hypothesis (GRH), every sufficiently large odd number was the sum of three primes. In 1937, Ivan Vinogradov removed the hypothesis of GRH and proved that every sufficiently large odd number n is the sum of three primes. In 1939 Vinogradov's student, K.G.Borozdkin, quantified the phrase sufficiently large, showing that would suffice. This bound has since been lowered a number of times, with the currently best known result due to Liu Ming-Chit and Wang Tian-Ze in 2002, who proved that every odd number is the sum of three primes. In principle, this leaves only a finite number of cases to check, but this is far too large a number to be handled by computer search (which, as mentioned earlier, has only reached as far as for the strong Goldbach conjecture, and not much further than that for the weak Goldbach conjecture). In 1997 Deshoulliers, Effinger, Te Riele, and Zinoviev were able to close the gap and prove that all odd numbers (greater than 5) are the sum of three primes, but only by assuming GRH again. In number theory, Goldbachs weak conjecture, also known as the odd Goldbach conjecture or the 3-primes problem, states that: Every odd number greater than 7 can be expressed as the sum of three odd primes. ... 1923 (MCMXXIII) was a common year starting on Monday (link will take you to calendar). ... G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ... John Edensor Littlewood (June 9, 1885 – September 6, 1977) was a British mathematician. ... The Riemann hypothesis is one of the most important conjectures in mathematics. ... In mathematics, the phrase sufficiently large is used in contexts such as: is true for sufficiently large which is actually shorthand for: there exists an such that is true for all . ... 1937 (MCMXXXVII) was a common year starting on Friday (link will take you to calendar). ... The neutrality of this article is disputed. ... 1939 (MCMXXXIX) was a common year starting on Sunday (link will take you to calendar). ... For album titles with the same name, see 2002 (album). ... 1997 (MCMXCVII) was a common year starting on Wednesday of the Gregorian calendar. ...

The strong Goldbach conjecture is much more difficult. The work of Vinogradov in 1937 and Theodor Estermann (1902-1991) in 1938 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, L.G. Schnirelmann proved that every even number n ≥ 4 can be written as the sum of at most 300,000 primes. This result was subsequently improved by many authors; currently, the best known result is due to Olivier Ramaré, who in 1995 showed that every even number n  ≥ 4 is in fact the sum of at most six primes. The neutrality of this article is disputed. ... 1937 (MCMXXXVII) was a common year starting on Friday (link will take you to calendar). ... Theodor Estermann (born 5 February 1902, Neubrandenburg; died 29 November 1991) was a mathematician, working in the field of analytic number theory. ... Year 1938 (MCMXXXVIII) was a common year starting on Saturday (link will take you to calendar). ... In mathematics, the phrase almost all has a number of specialised uses. ... 1930 (MCMXXX) was a common year starting on Wednesday (link is to a full 1930 calendar). ... Lev G. Schnirelmann was a Soviet mathematician who sought to prove Goldbachs conjecture. ... Olivier Ramaré is a French mathematician who teaches at the Université des Sciences et Technologies de Lille. ... 1995 (MCMXCV) was a common year starting on Sunday of the Gregorian calendar. ...

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)^[1]—e.g., 100 = 23 + 7·11. Chen Jingrun (ch. ... 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. ... In mathematics, the phrase sufficiently large is used in contexts such as: is true for sufficiently large which is actually shorthand for: there exists an such that is true for all . ... In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers. ...

In 1975, Hugh Montgomery and Robert Charles Vaughan showed that "most" even numbers were expressible as the sum of two primes. More precisely, they showed that there existed positive constants c,C such that for all sufficiently large numbers N, every even number less than N is the sum of two primes, with at most CN^{1 − c} exceptions. In particular, the set of even integers which are not the sum of two primes has density zero. 1975 (MCMLXXV) was a common year starting on Wednesday. ... Hugh Montgomery is an American mathematician, working in the fields of analytic number theory and mathematical analysis. ... Robert Charles (Bob) Vaughan (born 24 March 1945) is a British mathematician, working in the field of analytic number theory. ...

Roger Heath-Brown and Jan-Christoph Schlage-Puchta showed in 2002, that every sufficiently large even integer is a sum of two primes and exactly 13 powers of 2.^[2] (David Rodney) Roger Heath-Brown is a British mathematician, working in the field of analytic number theory. ...

One can pose similar questions when primes are replaced by other special sets of numbers, such as the squares. For instance, it was proven by Lagrange that every positive integer is the sum of four squares. See Waring's problem. Lagranges four-square theorem, also known as Bachets conjecture, was proved in 1770 by Joseph Louis Lagrange. ... In number theory, Warings problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers. ...

Trivia

• Doug Lenat's Automated Mathematician rediscovered Goldbach's Conjecture in 1982. This is considered one of the earliest demonstrations that artificial intelligences are capable of scientific discovery.
• To generate publicity for the book Uncle Petros and Goldbach's Conjecture by Apostolos Doxiadis, British publisher Tony Faber offered a $1,000,000 prize for a proof of the conjecture in 2000, if a proof was submitted before April 2002. The prize was never claimed.
• The television drama Lewis featured a mathematics professor at Oxford University who had won the Fields medal for his work on Goldbach's conjecture, which was a main plot feature.
• SF author Stephen Baxter penned a short story Planck Zero in which the Goldbach Conjecture features prominently.

Douglas B. Lenat is the CEO of Cycorp, Inc. ... The Automated Mathematician is one of the earliest successful discovery systems developed. ... Hondas humanoid robot AI redirects here. ... Uncle Petros and Goldbachs Conjecture is a 1992 novel by Greek author Apostolos Doxiadis. ... Apostolos Doxiadis (Greek: Απόστολος Δοξιάδης) (b. ... Lewis (known as Inspector Lewis in the US) is a British television drama made as a spin-off from Inspector Morse. ... The University of Oxford, located in the city of Oxford in England, is the oldest university in the English-speaking world. ... The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, a meeting that takes place every four years. ... Stephen Baxter at the Science-Fiction-Tage NRW in Dortmund, Germany, March 1997 Stephen Baxter (born in Liverpool, 13 November 1957) is a British hard science fiction author. ...

Attempted proofs

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

Because it is easily understood by laymen, Goldbach's conjecture is a popular target for pseudomathematicians who attempt to prove it, sometimes even disprove it, using only high-school-level mathematics. It shares this fate with the four-color theorem and Fermat's last theorem, each of which also has an easily stated problem, but a current proof which is extraordinarily elaborate. Pseudomathematics is a form of mathematics-like activity undertaken exclusively by non-mathematicians and having the distinguishing characteristic of producing no advancement in the field of rigorous mathematics (although work so produced may be of some aesthetic or artistic value). ... Example of a four color map The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same colour. ... Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats Last Theorem (edition of 1670). ...

It is possible that Goldbach's conjecture can yield to simple methods, but given the amount of professional attention paid to the conjecture, it is unlikely that a proof or a counter-example will be easy to find.

External links

• Goldbach's original letter to Euler - PDF format
• Goldbach's conjecture, part of Chris Caldwell's Prime Pages.
• A million-dollar maths question. Article by Anjana Ahuja in The Times, March 16, 2000.
• Goldbach conjecture verification, Tomás Oliveira e Silva's distributed computer search.
• Online tool to test Goldbach's conjecture on submitted integers.
• Goldbach Weave showing a graphical representation of Goldbach's conjecture.

The Prime pages is a website about prime numbers maintained by Prof. ... The Times is a national newspaper published daily in the United Kingdom since 1785, and under its current name since 1788. ...

See also

• Euler prime

In number theory, Euler primes or symmetric primes are primes that are the same distance from a given integer. ...

References

1. ^ J. R. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Sci. Sinica 16 (1973), 157--176.
2. ^ D. R. Heath-Brown, J. C. Puchta, Integers represented as a sum of primes and powers of two. The Asian Journal of Mathematics, 6 (2002), no. 3, pages 535-565.

• J.-M. Deshouillers; G. Effinger; H. te Riele; D. Zinoviev, A complete Vinogradov 3-primes theorem under the Riemann hypothesis, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 99--104 (electronic).
• Apostolos Doxiadis: Uncle Petros and Goldbach's Conjecture. ISBN 1-58234-128-1.
• H.L. Montgomery, Vaughan, R. C., The exceptional set in Goldbach's problem. Collection of articles in memory of Jurii Vladimiroviv Linnik. Acta Arith. 27 (1975), 353--370.