Conjectures
Goldbach Conjecture
German: [ˈɡɔltˌbax], approximately GOLT-bahkh (named for Christian Goldbach)
Also Known As Strong Goldbach Conjecture
Number Theory
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One of the oldest unsolved problems in number theory, and among the easiest to state. In a 1742 letter to Leonhard Euler, the Prussian mathematician Christian Goldbach (not yet a live entity on this atlas) proposed that every integer greater than two is the sum of three primes; Euler replied with the now-standard, equivalent, stronger form concerning even numbers. Every even number checked, into the many quintillions by computer search, satisfies the conjecture, and no exception has ever been found, but no general proof exists. Substantial partial results exist, including a 1937 proof by Ivan Vinogradov that every sufficiently large odd number is the sum of three primes.
Facts
StatementEvery even integer greater than two can be written as the sum of two prime numbers. 1 Proposed YearGoldbach's original 1742 letter to Euler proposed a related three-prime form; the modern two-prime, even-number form is the version Euler restated in his reply and the version now called the conjecture. Prize StatusNot a Millennium Prize Problem; no major institutional cash prize is offered for its proof (a commercial publisher briefly offered a one million dollar prize in 2000-2002 to promote a novel, since expired unclaimed). 1 Progress Toward ResolutionThe weaker ternary (odd) form, that every odd number greater than five is the sum of three primes, was proved unconditionally by Harald Helfgott in 2013, resolving a question open since Vinogradov's 1937 result for sufficiently large odd numbers. The original binary (even-number) conjecture itself remains unproven, verified by computer for every even number checked into the many quintillions with no exception found. 1 Cross-Tradition Connections
Associated With
Euler received Goldbach's original 1742 letter and restated the conjecture in its now-standard even-number form; Goldbach himself is not yet a live entity on this atlas.
In Branch
Posed By
Proposed in a 1742 letter to Euler; Euler's reply restated it in the modern even-number form.
Sources
1. MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and Statistics
Goldbach (Wiktionary)
Wikimedia FoundationGerman section, IPAQuote, German section, IPA
[ˈɡɔltˌbax]
Goldbach's Conjecture (Wikipedia)
Wikimedia FoundationPosed By: Christian Goldbach, Origins section, Goldbach's June 7 1742 letter to EulerQuote, Posed By: Christian Goldbach, Origins section, Goldbach's June 7 1742 letter to Euler
Every integer that can be written as the sum of two primes can also be written as the sum of as many primes (including unity) as one wishes, until all terms are units.
Goldbach's Conjecture (Wikipedia)
Wikimedia FoundationAssociated With: Leonhard Euler, Origins section, Euler's June 30 1742 replyQuote, Associated With: Leonhard Euler, Origins section, Euler's June 30 1742 reply
That ... every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it.
Goldbach's Conjecture (Wikipedia)
Wikimedia FoundationIn Branch: Number Theory, Opening paragraphQuote, In Branch: Number Theory, Opening paragraph
one of the oldest and best-known unsolved problems in number theory and all of mathematics
Open Questions (1 open question)
Can every even integer greater than two really always be written as the sum of two primes?
Verified by computer for every even number checked so far, into the many quintillions, with no exception found, but no general proof covering all even numbers exists, nearly three centuries after Goldbach first proposed it.
What would resolve this A general proof covering every even integer (or a single confirmed counterexample, an even number that is not the sum of two primes).
Analytic number theoryMacTutor History of Mathematics Archive
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