quality:correction, structural:reformatting, structural:expansion
Origins

On 7 June 1742, the Prussian 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 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

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.