Goldbach's conjecture

From Includipedia, the inclusionist encyclopedia

Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:

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

Expressing a given even number as a sum of two primes is called a Goldbach partition of the number. For example,

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.

[edit] 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.

He considered 1 to be a prime number, a convention subsequently abandoned. 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, answered by noting that this conjecture would follow from a stronger version,

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, which can be restated to the conjecture that all odd numbers greater than 7 are the sum of three odd primes, is known today as the "weak" Goldbach conjecture, the "odd" Goldbach conjecture, or the "ternary" Goldbach conjecture. The latter conjecture is known today as the "strong", "even", or "binary" 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. If the strong Goldbach conjecture is true, then it also implies that the weak Goldbach conjecture is true.[2]

[edit] 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.

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 <math>1/\ln m\,\!</math> 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 <math>1 \big / \big [\ln m \,\ln (n-m)\big ]</math>. This heuristic is non-rigorous for a number of reasons; for instance, it assumes that the events that m and <math>n-m</math> 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

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

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 <math>n-m</math> being prime. For instance, if m is odd then <math>n-m</math> is also odd, and if m is even, then <math>n-m</math> 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 <math>n-m</math> 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 <math>n=p_1+\dotsb+p_c</math> with <math>p_1 \leq \dotsb \leq p_c</math> should be asymptotically equal to

<math> \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}</math>

where the product is over all primes p, and <math>\gamma_{c,p}(n)</math> is the number of solutions to the equation <math>n = q_1 + \ldots + q_c \mod p</math> in modular arithmetic, subject to the constraints <math>q_1,\ldots,q_c \neq 0 \mod p</math>. 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 <math>c=2</math>. In the latter case, the above formula simplifies to 0 when n is odd, and to

<math> 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} </math>

when n is even, where <math>\Pi_2</math> is the twin prime constant

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

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.

[edit] 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 <math>n \leq 10^5</math>. 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 <math>n \leq 10^{18}</math> (as of April 2007).

The weak Goldbach conjecture is fairly close to resolution.

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, Lev 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. 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)^[3]—e.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. More precisely, they showed that there existed positive constants <math>c, C</math> such that for all sufficiently large numbers N, every even number less than N is the sum of two primes, with at most <math>C N^{1-c}</math> exceptions. In particular, the set of even integers which are not the sum of two primes has density zero.

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.^[4]

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.

[edit] 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, sometimes 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.

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

[edit] 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. 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.
• Isaac Asimov's short story "Sixty Million Trillion Combinations" featured a mathematician who suspected that his work on Goldbach's conjecture had been stolen.

[edit] References

[edit] Further reading

• 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 Vladimirovich Linnik. Acta Arith. 27 (1975), 353--370.

[edit] External links

• Goldbach's original letter to Euler - PDF format (in German)
• 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.
• 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.ar:حدسية غولدباخ

bn:গোল্ডবাখ অনুমান zh-min-nan:Goldbach Chhai-chhek ca:Conjectura de Goldbach cs:Goldbachova hypotéza da:Goldbachs formodning de:Goldbachsche Vermutung el:Εικασία του Γκόλντμπαχ es:Conjetura de Goldbach eo:Konjekto de Goldbach fa:حدس گلدباخ fr:Conjecture de Goldbach zh-classical:哥德巴赫猜想 ko:골트바흐의 추측 id:Konjektur Goldbach it:Congettura di Goldbach he:השערת גולדבך hu:Goldbach-sejtés nl:Vermoeden van Goldbach ja:ゴールドバッハの予想 no:Goldbachs formodning pms:Congetura ëd Goldbach pl:Hipoteza Goldbacha pt:Conjectura de Goldbach ro:Conjectura lui Goldbach ru:Проблема Гольдбаха scn:Cungittura di Goldbach sl:Goldbachova domneva fi:Goldbachin konjektuuri sv:Goldbachs hypotes th:ข้อความคาดการณ์ของโกลด์บาช tr:Goldbach hipotezi zh:哥德巴赫猜想