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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).
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 conjecture: number theory: ⇒The ternary Goldbach conjecture, which was the original formulation. [8] Christian Goldbach: 5880 Gold partition conjecture [9] order theory: n/a: 25 Goldberg–Seymour conjecture: graph theory: Mark K. Goldberg and Paul Seymour: 57 Goormaghtigh conjecture: number theory: René Goormaghtigh: 14 Green's ...
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]
However, based on the different meaning of the words in constructive mathematics, if there is a constructive proof that "α = 0 or α ≠ 0" then this would mean that there is a constructive proof of Goldbach's conjecture (in the former case) or a constructive proof that Goldbach's conjecture is false (in the latter case).
Henry Andrew Pogorzelski (September 26, 1922 - December 30, 2015) [1] was an American mathematician of Polish descent, [2] a professor of mathematics at the University of Maine. Much of Pogorzelski's research concerns the Goldbach conjecture , the still-unsolved problem of whether every even number can be represented as a sum of two prime numbers .
Goldbach's conjecture, one of the oldest unsolved problems in number theory; Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem; Goldbach's comet, a plot of the so-called Goldbach function; Goldbach–Euler theorem, also known as Goldbach's theorem
The arithmetical example above provides what is called a weak counterexample. The existence claim ∃x (φ → θ) cannot be provable by intuitionistic means: Being able to inspect an x validating φ → θ would resolve the conjecture. For example, consider the following classical argument: Either the Goldbach conjecture has a proof or it does ...