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This conjecture is called "weak" because if Goldbach's strong conjecture (concerning sums of two primes) is proven, then this would also be true. For if every even number greater than 4 is the sum of two odd primes, adding 3 to each even number greater than 4 will produce the odd numbers greater than 7 (and 7 itself is equal to 2+2+3).
The best known result currently stems from the proof of the weak Goldbach conjecture by Harald Helfgott, [15] which directly implies that every even number n ≥ 4 is the sum of at most 4 primes. [ 16 ] [ 17 ]
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.
Proof of Goldbach's weak conjecture: Awards: Leverhulme Prize (2008) Whitehead Prize (2010) Adams Prize (2011) Humboldt Professorship (2015) Scientific career: Fields: Mathematics: Institutions: CNRS/Institut de mathématiques de Jussieu University of Göttingen: Doctoral advisor: Henryk Iwaniec [1] [2] Peter Sarnak [2]
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.
"There exists a natural number x, such that if an index of a proof of the Goldbach conjecture exists, then the number x is the index of a proof of the Goldbach conjecture." This is classically provable, as follows: Either an index for a proof of the Goldbach conjecture exists, or no such index exists.
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This result may be compared with Goldbach's conjecture, which states that every even number except 2 is the sum of two primes. The truth of Ramaré's result for all sufficiently large even numbers is a consequence of Vinogradov's theorem, whereas the full result follows from Goldbach's weak conjecture.