Goldbach's conjecture

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Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:

Expressing a given even number as a sum of two primes is called a Goldbach partition of the number. For example,

Contents

Origins

On 7 June 1742, the German mathematician Christian Goldbach of originally Brandenburg-Prussia wrote a letter to Leonhard Euler (letter XLIII)[3] in which he proposed the following conjecture:

He then proposed a second conjecture in the margin of his letter:

He considered 1 to be a prime number, a convention subsequently abandoned.[4] The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time. A modern version of Goldbach's marginal conjecture is:

Euler replied in a letter dated 30 June 1742, and reminded Goldbach of an earlier conversation they had ("...so Ew vormals mit mir communicirt haben.."), in which Goldbach remarked his original (and not marginal) conjecture followed from the following statement

which is thus also a conjecture of Goldbach. 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 necht demonstriren kann.” ("every even integer is a sum of two primes. I regard this as a completely certain theorem, although I cannot prove it.")[5][6]

Goldbach's third version (equivalent to the two other versions) 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.[6]

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