Goldbach Conjecture

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That every even integer greater than 3 can be written as the sum of two

Also known as: The “strong”, “even” or “binary” Goldbach conjecture
because it implies the “weak”, “odd” or “ternary” Goldbach conjecture that
all odd numbers greater than seven are the sum of three odd primes. The
conjecture does not specify that a number has to be the sum of only one
pair of prime numbers.

Discipline: Number theory.

Originator: Prussian mathematician Christian Goldbach wrote to Leonhard
Euler in 1742 proposing that every integer greater than two can be written
as the sum of three primes. Euler replied that it follows that every even
integer greater than two can be written as the sum of two primes. Euler’s form
has since been known by Goldbach’s name.

Incentive: Proving one of the oldest unsolved problems in number theory and
all mathematics. A $1million prize offered by publisher Faber & Faber for a proof
submitted before April 2002 was never claimed.

Examples: 4 = 2+2, 6 = 3+3, 8 = 3+5, 10 = 5+5 = 3+7, 12 = 5+7, and so on.

Verified results: For smaller numbers, the strong Goldbach conjecture can be directly
verified. One 1938 attempt laboriously verified up to n ≤ 105, while a distributed
computer search has verified the conjecture for n ≤ 1018.

State of play: There is little doubt among mathematicians that both conjectures are
true; Euler replied to Goldbach: “That every even number is a sum of two primes,
I consider an entirely certain theorem in spite of that I am not able to demonstrate
it.” No purported proofs are currently accepted by the mathematical community.

The weak Goldbach conjecture is fairly close to resolution, but the strong
conjecture is much harder to verify. It has been shown that every even number n ≥ 4
is the sum of at most six primes.

Statistical work on the probabilistic distribution of prime numbers presents
informal evidence for the conjecture for sufficiently large integers.

Published in IMAges 6 - April 2009