Goldbach's conjecture

Formal statement

Each of the three conjectures 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:

Every integer that can be written as the sum of two primes can also be written as the sum of as many primes as one wishes, until either all terms are two (if the integer is even) or one term is three and all other terms are two (if the integer is odd).

A modern version of the marginal conjecture is:

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

And a modern version of Goldbach's older conjecture of which Euler reminded him is:

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

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

Every odd integer greater than 7 can be written as the sum of three odd primes.

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/ln m ln(n − m)⁠. 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

$${\displaystyle \sum _{m=3}^{\frac{n}{2}}\frac{1}{\ln m}\frac{1}{\ln (n-m)}\approx \frac{n}{2(\ln n{)}^2}.}$$

Since ${\displaystyle \ln n\ll \sqrt{n}}$, 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 other than 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 = p₁ + ⋯ + p_c with p₁ ≤ ⋯ ≤ p_c should be asymptotically equal to

$${\displaystyle \left(\prod _p\frac{p{\gamma }_{c,p}(n)}{(p-1{)}^c}\right){\int }_{2\le x_1\le \cdots \le x_c:x_1+\cdots +x_c=n}\frac{d{x}_1\cdots d{x}_{c-1}}{\ln x_1\cdots \ln x_c},}$$

where the product is over all primes p, and γ_c,p(n) is the number of solutions to the equation n = q₁ + ⋯ + q_c mod p in modular arithmetic, subject to the constraints q₁, ..., q_c ≠ 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

$${\displaystyle 2{\mathrm{\Pi }}_2\left(\prod _{p\mid n;p\ge 3}\frac{p-1}{p-2}\right){\int }_2^n\frac{dx}{(\ln x{)}^2}\approx 2{\mathrm{\Pi }}_2\left(\prod _{p\mid n;p\ge 3}\frac{p-1}{p-2}\right)\frac{n}{(\ln n{)}^2}}$$

when n is even, where Π₂ is Hardy–Littlewood's twin prime constant

$${\displaystyle {\mathrm{\Pi }}_2:=\prod _{p\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\ge 3}\left(1-\frac{1}{(p-1{)}^2}\right)\approx 0.660161815846869573927812110014\dots }$$

This is sometimes known as the extended Goldbach conjecture. The Goldbach conjecture is in fact very similar to the twin prime conjecture, and the two conjectures are believed to be of roughly comparable difficulty.

Goldbach partition function

The Goldbach partition function associates to each even integer the number of ways it can be decomposed into a sum of two primes. Its graph looks like a comet and is therefore called Goldbach's comet.

Goldbach's comet suggests tight upper and lower bounds on the number of representations of an even number as the sum of two primes, and also that the number of these representations depends strongly on the value modulo 3 of the number.

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:

• Lagrange proved that every positive integer is the sum of four squares. See Waring's problem and the related Waring–Goldbach problem on sums of powers of primes.
• Hardy and Littlewood listed as their Conjecture I: "Every large odd number (n > 5) is the sum of a prime and the double of a prime". This conjecture is known as Lemoine's conjecture and is also called Levy's conjecture.
• The Goldbach conjecture for practical numbers, a prime-like sequence of integers, was stated by Margenstern in 1984, and proved by Melfi in 1996: every even number is a sum of two practical numbers.
• Harvey Dubner proposed a strengthening of the Goldbach conjecture: that every even integer greater than 4208 is the sum of two twin primes (not necessarily belonging to the same pair). Only 34 even integers less than 4208 are not the sum of two twin primes; Dubner has verified computationally that this list is complete up to ${\displaystyle 2\cdot {10}^{10}.}$ A proof of this stronger conjecture would imply not only Goldbach's conjecture but also the twin prime conjecture.

Goldbach's conjecture is used to study computational complexity. The connection is made through the Busy Beaver function, where BB(n) is the maximum number of steps taken by any n state Turing machine that halts. There is a 27-state Turing machine that halts if and only if Goldbach's conjecture is false. Hence if BB(27) was known, and the Turing machine did not stop in that number of steps, it would be known to run forever and hence no counterexamples would exist (proving the conjecture). This is a completely impractical way to settle the conjecture; instead it is used to suggest that BB(27) will be very hard to compute, at least as difficult as settling the Goldbach conjecture.

In popular culture

Goldbach's Conjecture (Chinese: 哥德巴赫猜想) is the title of Xu Chi's biography of Chinese mathematician and number theorist Chen Jingrun.

The conjecture is a central point in the plot of Apostolos Doxiadis's 1992 novel Uncle Petros and Goldbach's Conjecture, in Isaac Asimov's short story "Sixty Million Trillion Combinations", and in Michelle Richmond's 2008 mystery novel No One You Know.

Goldbach's conjecture is part of the plot of the 2007 Spanish film Fermat's Room.

Goldbach's conjecture is featured as the main topic of research of the eponymous character Marguerite in the 2023 French-Swiss film Marguerite's Theorem.

Goldbach's conjecture is a plot device in the Frederik Pohl novella "The Gold at the Starbow's End."