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 $4\times 10^{1}8$, but remains unproven despite considerable effort.

1 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:

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 all terms are units.

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:

… eine jede Zahl, die grösser ist als 2, ein aggregatum trium numerorum primorum sey.
Every integer greater than $2$ can be written as the sum of three primes.

Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had had (“… so Ew vormals mit mir communicirt haben …”), in which Goldbach had remarked that the first of those two conjectures would follow from the statement

Every positive even integer can be written as the sum of two primes.

This is in fact equivalent to his second, marginal conjecture. 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.
That … every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it.

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:

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.

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 $n–3$ is a sum of two primes, then $n$ is a sum of three primes. However, the converse implication and thus the strong Goldbach conjecture remain unproven.

2 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 $n\le 10^5$. 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 $n\le 4\times 10^{1}8$ (and double-checked up to $4\times 10^{1}7$) as of 2013. One record from this search is that 3325581707333960528 is the smallest number that cannot be written as a sum of two primes where one is smaller than 9781.

3 Heuristic justification

4 Rigorous results

5 Related problems

6 In popular culture

7 References

8 Further reading

9 External links