Goldbach's conjecture

...On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) [2] 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.^[3] 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, replied by noting that this conjecture is equivalent to another version:

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

adding that he regarded this an entirely certain theorem ("ein ganz gewisses Theorema"), despite being unable to prove it.^[4]

Euler's 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 corollary. The strong Goldbach conjecture implies the conjecture that all odd numbers greater than 7 are the sum of three odd primes, which is known today variously as the "weak" Goldbach conjecture, the "odd" Goldbach conjecture, or the "ternary" Goldbach conjecture. Both questions have remained unsolved ever since, although the weak form of the conjecture appears to be much closer to resolution than the strong one. If the strong Goldbach conjecture is true, the weak Goldbach conjecture will be true by implication.[3] ....