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In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, is the now-proven statement 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).
This is sometimes known as the extended Goldbach conjecture. The strong Goldbach conjecture is in fact very similar to the twin prime conjecture, and the two conjectures are believed to be of roughly comparable difficulty. Goldbach's comet; red, blue and green points correspond respectively the values 0, 1 and 2 modulo 3 of the number.
Goldbach's weak conjecture, every odd number greater than 5 can be expressed as the sum of three primes, is a consequence of Goldbach's conjecture. Ivan Vinogradov proved it for large enough n (Vinogradov's theorem) in 1937, [1] and Harald Helfgott extended this to a full proof of Goldbach's weak conjecture in 2013. [2] [3] [4]
Uniformity conjecture: diophantine geometry: n/a: Unique games conjecture: number theory: n/a: Vandiver's conjecture: number theory: Ernst Kummer and Harry Vandiver: Virasoro conjecture: algebraic geometry: Miguel Ángel Virasoro: Vizing's conjecture: graph theory: Vadim G. Vizing: Vojta's conjecture: number theory: ⇒abc conjecture: Paul ...
Goldbach's weak conjecture also follows from the generalized Riemann hypothesis. The yet to be verified proof of Harald Helfgott of this conjecture verifies the GRH for several thousand small characters up to a certain imaginary part to obtain sufficient bounds that prove the conjecture for all integers above 10 29 , integers below which have ...
The Waring–Goldbach problem is a problem in additive number theory, concerning the representation of integers as sums of powers of prime numbers. It is named as a combination of Waring's problem on sums of powers of integers, and the Goldbach conjecture on sums of primes. It was initiated by Hua Luogeng [1] in 1938.
Chen's theorem is a significant step towards Goldbach's conjecture, and a celebrated application of sieve methods. Chen's theorem represents the strengthening of a previous result due to Alfréd Rényi , who in 1947 had shown there exists a finite K such that any even number can be written as the sum of a prime number and the product of at most ...
In number theory, Vinogradov's theorem is a result which implies that any sufficiently large odd integer can be written as a sum of three prime numbers.It is a weaker form of Goldbach's weak conjecture, which would imply the existence of such a representation for all odd integers greater than five.