Goldbach's conjecture


Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states:
The conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort.

Goldbach number

A Goldbach number is a positive even integer that can be expressed as the sum of two odd primes. Since 4 is the only even number greater than 2 that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers.
The expression of a given even number as a sum of two primes is called a Goldbach partition of that number. The following are examples of Goldbach partitions for some even numbers:
The number of ways in which 2n can be written as the sum of two primes is:

Origins

On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler, in which he proposed the following conjecture:
He then proposed a second conjecture in the margin of his letter:
He considered 1 to be a prime number, a convention subsequently abandoned. The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time.
A modern version of Goldbach's marginal conjecture is:
Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had, in which Goldbach remarked his original conjecture followed from the following statement
which is, thus, also a conjecture of Goldbach.
In the letter dated 30 June 1742, Euler stated:

"Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann."

Goldbach's third 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 conjecture, known today variously as the Goldbach's weak conjecture, the "odd" Goldbach conjecture, or the "ternary" Goldbach conjecture. This weak conjecture asserts that all odd numbers greater than 7 are the sum of three odd primes and appears to have been proved in 2013. The weak conjecture is a corollary of the strong conjecture: if is a sum of two primes, then is a sum of three primes. The converse implication and the strong Goldbach conjecture remain unproven.

Verified results

For small values of n, the strong Goldbach conjecture can be verified directly. For instance, Nils Pipping in 1938 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 as of 2013. One record from this search is that is the smallest number that has no Goldbach partition with a prime below 9781.

Heuristic justification

Statistical considerations that focus on the probabilistic distribution of prime numbers present informal evidence in favour of the conjecture 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 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. 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
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.
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, Hardy and Littlewood in 1923 conjectured 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
where the product is over all primes p, and 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. In the latter case, the above formula simplifies to 0 when n is odd, and to
when n is even, where is Hardy–Littlewood's twin prime constant
This 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.
The Goldbach partition functions shown here can be displayed as histograms, which informatively illustrate the above equations. See Goldbach's comet.

Rigorous results

The strong Goldbach conjecture is much more difficult than the weak Goldbach conjecture. Using Vinogradov's method, Chudakov, Van der Corput, and Estermann showed that almost all even numbers can be written as the sum of two primes. In 1930, Lev Schnirelmann proved that any natural number 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 amount of even numbers up to violating the Goldbach conjecture is much less than for small.
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. See Chen's theorem 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 exceptions. In particular, the set of even integers that are not the sum of two primes has density zero.
In 1951, 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 and Jan-Christoph Schlage-Puchta in 2002 found that works.
As with many famous conjectures in mathematics, there are a number of purported proofs of the Goldbach conjecture, none of which are accepted by the mathematical community.

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.
One can consider similar problems in which primes are replaced by other particular sets of numbers, such as the squares.
Goldbach's Conjecture is the title of the biography of Chinese mathematician and number theorist Chen Jingrun, 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 and also in the 2008 mystery novel No One You Know by Michelle Richmond.
Goldbach's conjecture is part of the plot of the Spanish film "Fermat's Room" .