Goldbach\'s conjecture

Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. This article lists some unsolved problems in Mathematics. See individual articles for details and sources 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 Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and It states:

Every even integer greater than 2 can be written as the sum of two primes. In Mathematics, the parity of an object states whether it is even or odd The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

Expressing a given even number as a sum of two primes is called a Goldbach partition of the number. In Number theory, a partition of a positive Integer n is a way of writing n as a Sum of positive integers For example,

4 = 2 + 2

6 = 3 + 3

8 = 3 + 5

10 = 3 + 7 = 5 + 5

12 = 5 + 7

14 = 3 + 11 = 7 + 7

…

In other words, the Goldbach conjecture states that every even number greater than or equal to four is a Goldbach number, a number that can be expressed as the sum of two primes. ^[1] See also Levy's conjecture.

An illustration of Goldbach's conjecture. ^[2]

1 Origins
2 Verified results
3 Heuristic justification
4 Rigorous results
5 Attempted proofs
6 In popular culture
7 References
8 Further reading
9 External links

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:

Every integer greater than 2 can be written as the sum of three primes. Events 1099 - The First Crusade: The Siege of Jerusalem begins Year 1742 ( MDCCXLII) was a Common year starting on Monday (link will display the full calendar of the Gregorian calendar (or a The Kingdom of Prussia (Königreich Preußen was a German kingdom from 1701 to 1918 and from 1871 was the leading state of the German Empire, comprising A mathematician is a person whose primary area of study and research is the field of Mathematics. Christian Goldbach ( March 18, 1690 &ndash November 20, 1764) was a Prussian Mathematician who also studied Law

He considered 1 to be a prime number, a convention subsequently abandoned. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 A modern version of Goldbach's original conjecture is:

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

Euler, becoming interested in the problem, replied by noting that this conjecture is equivalent with another version:

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

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

Euler's version is the form in which the conjecture is usually expressed today. It is also known as the "strong", "even", or "binary" Goldbach conjecture, to distinguish it from a weaker corollary. The Language of mathematics has a vast Vocabulary of specialist and technical terms The strong Goldbach conjecture implies the conjecture that all odd numbers greater than 7 are the sum of three odd primes, which is known today variously as the "weak" Goldbach conjecture, the "odd" Goldbach conjecture, or the "ternary" 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 Both questions have remained unsolved ever since, although the weak form of the conjecture appears to be much closer to resolution than the strong one. If the strong Goldbach conjecture is true, the weak Goldbach conjecture will be true by implication. [2]

Verified 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 $n \leq 10^5$ . 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 for $n \leq 10^{18}$ ^[3].

The Goldbach conjecture does not say that a number must be the sum of a unique pair of prime numbers. The examples in this article illustrate that more than one pair of prime numbers may sum to the same number.

Heuristic justification

Statistical considerations which 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. In Mathematics, the phrase sufficiently large is used in contexts such as P is true for sufficiently large x which is actually shorthand

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. heuristic (hyu̇-ˈris-tik is a method to help solve a problem commonly an informal method The prime number theorem 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 $1 \big / \big [\ln m \,\ln (n-m)\big ]$ . 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. 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 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

$\sum_{m=3}^{n/2} \frac{1}{\ln m} {1 \over \ln (n-m)} \approx \frac{n}{2 \ln^2 n}.$

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 dependence between the events 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. In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than 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 $n=p_1+\dotsb+p_c$ with $p_1 \leq \dotsb \leq p_c$ should be asymptotically equal to

$\left(\prod_p \frac{p \gamma_{c,p}(n)}{(p-1)^c}\right) \int_{2 \leq x_1 \leq \dotsb \leq x_c: x_1+\ldots+x_c = n} \frac{dx_1 \ldots dx_{c-1}}{\ln x_1 \ldots \ln x_c}$

where the product is over all primes p, and $γ c, p (n)$ is the number of solutions to the equation $n = q_1 + \ldots + q_c \mod p$ in modular arithmetic, subject to the constraints $q_1,\ldots,q_c \neq 0 \mod p$ . Godfrey Harold Hardy FRS ( February 7, 1877 Cranleigh, Surrey, England &ndash December 1, 1947 John Edensor Littlewood ( 9 June 1885 &ndash 6 September 1977) was a British Mathematician, best known for his long collaboration In pure and Applied mathematics, particularly the Analysis of algorithms, real analysis and engineering asymptotic analysis is a method of describing In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers 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 $c = 2$ . Ivan Matveevich Vinogradov (Иван Матвеевич Виноградов September 14, 1891 &ndash March 20, 1983) was a Russian In the latter case, the above formula simplifies to 0 when n is odd, and to

$2 \Pi_2 \left(\prod_{p|n; p \geq 3} \frac{p-1}{p-2}\right) \int_2^n \frac{dx}{\ln^2 x} \approx 2 \Pi_2 \left(\prod_{p|n; p \geq 3} \frac{p-1}{p-2}\right) \frac{n}{\ln^2 n}$

when n is even, where $Π 2$ is the twin prime constant

$\Pi_2 := \prod_{p \geq 3} \left(1 - \frac{1}{(p-1)^2}\right) = 0.6601618158\ldots.$

This asymptotic is sometimes known as the extended Goldbach conjecture. The twin prime conjecture is a famous unsolved problem in Number theory that involves Prime numbers It states There are infinitely many primes In pure and Applied mathematics, particularly the Analysis of algorithms, real analysis and engineering asymptotic analysis is a method of describing 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. The twin prime conjecture is a famous unsolved problem in Number theory that involves Prime numbers It states There are infinitely many primes

The partition functions shown here can be displayed as histograms which informatively illustrate the above equations. See Goldbach's comet. Goldbach's comet is the name given to a plot of the function g(n the so-called Goldbach function.

Rigorous results

The weak Goldbach conjecture is fairly close to resolution. In Number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem

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). Ivan Matveevich Vinogradov (Иван Матвеевич Виноградов September 14, 1891 &ndash March 20, 1983) was a Russian Theodor Estermann ( 5 February 1902 &ndash 29 November[[ 991]] was a mathematician working in the field of Analytic number theory. See also Generic property In Mathematics, the phrase almost all has a number of specialised uses In 1930, Lev Schnirelmann proved that every even number n ≥ 4 can be written as the sum of at most 300,000 primes. Lev Genrikhovich Schnirelmann (Лев Генрихович Шнирельман also Shnirelman, Shnirel'man (born January 2, 1905 in 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. Olivier Ramaré is a French Mathematician who teaches at the Université des Sciences et Technologies de Lille. In fact, resolving the weak Goldbach conjecture will also directly imply that every even number n ≥ 4 is the sum of at most four primes.

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)^[4]—e. Chen Jingrun ( May 22 1933 – March 19 1996) was a Chinese Mathematician who made significant contributions to Number 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 P is true for sufficiently large x which is actually shorthand In Mathematics, a semiprime (also called biprime or 2- Almost prime, or pq number) is a Natural number that is the product g. , 100 = 23 + 7·11.

In 1975, Hugh Montgomery and Robert Charles Vaughan showed that "most" even numbers were expressible as the sum of two primes. Hugh Montgomery can be Hugh Montgomery 1st Viscount of the Great Ardes Hugh Montgomery (mathematician, an American mathematician 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 $C N 1 - c$ exceptions. In particular, the set of even integers which are not the sum of two primes has density zero. In Number theory, asymptotic density or natural density is one of the possibilities to measure how large is a Subset of the set of Natural

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. David Rodney ("Roger" Heath-Brown, FRS, is a British mathematician working in the field of Analytic number theory. ^[5]

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. Lagrange's four-square theorem, also known as Bachet's conjecture, was proven in 1770 by Joseph Louis Lagrange. See Waring's problem. In Number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every Natural number k there exists an associated positive

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 amateur mathematicians, who often attempt to prove or disprove it using only high-school-level mathematics. It shares this fate with the four-color theorem and Fermat's last theorem, both of which also have an easily stated problem but nevertheless appear to be solvable only through extraordinarily elaborate methods. The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like

In popular culture

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. Uncle Petros and Goldbach's Conjecture is a 1992 novel by Greek author Apostolos Doxiadis. 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. Lewis (known as Inspector Lewis in the United States) is a British television detective drama made as a spin-off from The University of Oxford (informally "Oxford University" or simply "Oxford" located in the city of Oxford, Oxfordshire, England is the 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
Isaac Asimov's short story "Sixty Million Trillion Combinations" featured a mathematician who suspected that his work on Goldbach's conjecture had been stolen.
In the Spanish movie "La habitación de Fermat" (2007), directed by Luis Piedrahita and Rodrigo Sopeña, a young mathematician claims to have solved the Conjecture. The art of motion-picture making within the nation of Spain or by Spanish filmmakers Luis Piedrahita (born 1977 in Coruña, Spain) is a Spanish stand-up comedian, magician script writer broadcaster and author
In the 2008 Futurama movie The Beast with a Billion Backs, Professors Farnsworth and Wernstrom collaborate on "yet another elementary proof of the Goldbach Conjecture" using fictional numbers such as Martian Primes. Futurama is an Emmy Award -winning animated American sitcom created by Matt Groening, and developed by Groening and Futurama The Beast with a Billion Backs is an animated Science-fiction Comedy, the second of the four Futurama straight-to-DVD

References

^ Eric W. Weisstein, Goldbach Number at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
^ “Goldbach's Conjecture" by Hector Zenil, The Wolfram Demonstrations Project, 2007.
^ Tomás Oliveira e Silva, http://www.ieeta.pt/~tos/goldbach.html, accessed on 25 April 2008
^ 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.
^ 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.

External links

Goldbach's original letter to Euler - PDF format (in German)
Goldbach's conjecture, part of Chris Caldwell's Prime Pages. The Prime Pages is a website about Prime numbers maintained by Prof
A million-dollar maths question. Article by Anjana Ahuja in The Times, March 16, 2000. Anjana Ahuja is a British Indian Science journalist and Columnist for The Times. The Times is a daily national Newspaper published in the United Kingdom since 1785 when it was known as The Daily Universal Register.
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.
Elementary proof of a very weak form of Goldbach's conjecture proves that the first N odd primes give at least 2N-1 unique even integers.

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