Goldbach conjecture

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.

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 wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture:

Goldbach was following the now-abandoned convention of considering 1 to be a prime number, 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", the "odd Goldbach conjecture", or the "ternary Goldbach conjecture", asserts that

A proof for the weak conjecture was proposed in 2013 by Harald Helfgott. 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 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 $1/\ln m$ 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 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, G. H. Hardy and John Edensor Littlewood 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 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, 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, 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 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 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.

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. Using Vinogradov's method, Nikolai Chudakov, Johannes van der Corput, and Theodor Estermann showed that almost all 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 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 number of even numbers up to violating the Goldbach conjecture is much less than $X^$ for small .

In 1948, using sieve theory, Alfréd Rényi 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 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). 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 $C N^$ exceptions. In particular, the set of even integers that are not the sum of two primes has density zero.

In 1951, Yuri 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 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 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.

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

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