Number Theory


Updated Elementary Proof of the Goldbach Conjecture

Authors: Stephen Marshall

Christian Goldbach (March 18, 1690 – November 20, 1764) was a German mathematician. He is remembered today for Goldbach's conjecture. Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every even integer greater than 2 can be expressed as the sum of two primes. On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) in which he proposed the following conjecture: Every even integer which can be written as the sum of two primes (the strong conjecture) He then proposed a second conjecture in the margin of his letter: Every odd integer greater than 7 can be written as the sum of three primes (the weak conjecture). A Goldbach number is a positive even integer that can be expressed as the sum of two odd primes. Since four is the only even number greater than two that requires the even prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all even integers greater than 4 are Goldbach numbers. The “strong” conjecture has been shown to hold up through 4 × 1018, but remains unproven for almost 300 years despite considerable effort by many mathematicians throughout history. In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that Every odd number greater than 5 can be expressed as the sum of three primes. (A prime may be used more than once in the same sum). In 2013, Harald Helfgott proved Goldbach's weak conjecture. The author would like to give many thanks to Helfgott’s proof of the weak conjecture, because this proof of the strong conjecture is completely dependent on Helfgott’s proof. Without Helfgott’s proof, this elementary proof would not be possible.

Comments: 6 Pages.

Download: PDF

Submission history

[v1] 2018-05-08 16:11:51
[v2] 2018-05-09 11:30:03

Unique-IP document downloads: 132 times is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus