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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 that every even natural number greater than 2 is the sum of two prime numbers.

Field

Number theory

Christian Goldbach

1742

Yes

Goldbach's weak conjecture

The conjecture has been shown to hold for all integers less than 4×1018 but remains unproven despite considerable effort.

History[edit]

Origins[edit]

On 7 June 1742, the Prussian mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII),[2] in which he proposed the following conjecture:

It was proven by that every positive integer is the sum of four squares. See Waring's problem and the related Waring–Goldbach problem on sums of powers of primes.

Lagrange

Hardy and Littlewood listed as their Conjecture I: "Every large odd number (n > 5) is the sum of a prime and the double of a prime". This conjecture is known as Lemoine's conjecture and is also called Levy's conjecture.

[31]

The Goldbach conjecture for , a prime-like sequence of integers, was stated by Margenstern in 1984,[32] and proved by Melfi in 1996:[33] every even number is a sum of two practical numbers.

practical numbers

proposed a strengthening of the Goldbach conjecture that states that every even integer greater than 4208 is the sum of two twin primes (not necessarily belonging to the same pair).[34] Only 34 even integers less than 4208 are not the sum of two twin primes; Dubner has verified computationally that this list is complete up to [35] A proof of this stronger conjecture would not only imply Goldbach's conjecture, but also the twin prime conjecture.

Harvey Dubner

Although Goldbach's conjecture implies that every positive integer greater than one can be written as a sum of at most three primes, it is not always possible to find such a sum using a greedy algorithm that uses the largest possible prime at each step. The Pillai sequence tracks the numbers requiring the largest number of primes in their greedy representations.[30]


Similar problems to Goldbach's conjecture exist in which primes are replaced by other particular sets of numbers, such as the squares:

Deshouillers, J.-M.; Effinger, G.; te Riele, H.; Zinoviev, D. (1997). (PDF). Electronic Research Announcements of the American Mathematical Society. 3 (15): 99–104. doi:10.1090/S1079-6762-97-00031-0.

"A complete Vinogradov 3-primes theorem under the Riemann hypothesis"

Montgomery, H. L.; Vaughan, R. C. (1975). (PDF). Acta Arithmetica. 27: 353–370. doi:10.4064/aa-27-1-353-370.

"The exceptional set in Goldbach's problem"

.

Terence Tao proved that all odd numbers are at most the sum of five primes

at MathWorld.

Goldbach Conjecture

Media related to Goldbach's conjecture at Wikimedia Commons

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Goldbach problem"

Goldbach's original letter to Euler — PDF format (in German and Latin)

part of Chris Caldwell's Prime Pages.

Goldbach's conjecture

Tomás Oliveira e Silva's distributed computer search.

Goldbach conjecture verification