Origins
On 7 June 1742, the German mathematician Christian Goldbach (originally of Brandenburg-Prussia) wrote a letter to Leonhard Euler (letter XLIII) in which he proposed the following conjecture:
- Every integer which 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.
He then proposed a second conjecture in the margin of his letter:
- 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. 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:
- Every integer greater than 5 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 ("…so Ew vormals mit mir communicirt haben…"), in which Goldbach remarked his original (and not marginal) conjecture followed from the following statement
- Every even integer greater than 2 can be written as the sum of two primes,
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.")
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.
Read more about this topic: Goldbach's Conjecture
Famous quotes containing the word origins:
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