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 larger than 4, for 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 , could not possibly rule out the existence of such a specific counterexample ). 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