In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that: Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...

Every odd number greater than 7 can be expressed as the sum of three odd primes.

(A prime may be used more than once in the same sum.) In mathematics, any integer (whole number) is either even or odd. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ...

This conjecture is called "weak" because Goldbach's strong conjecture concerning sums of two primes, if proven, would establish Goldbach's weak conjecture. (Since if every even number greater than 4 is the sum of two odd primes, merely adding three to each even number greater than 4 will produce the odd numbers greater than 7.) In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ... Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ...

The conjecture has not yet been proven, but there have been some helpful near misses. In 1923, Hardy and Littlewood showed that, assuming the generalized Riemann hypothesis, the odd Goldbach conjecture is true for all sufficiently large odd numbers. In 1937, a Russian mathematician, Ivan Matveevich Vinogradov, was able to eliminate the dependency on the Riemann hypothesis and proved directly (see Vinogradov's theorem) that all sufficiently large odd numbers can be expressed as the sum of three primes. Although Vinogradov himself was unable to specify "sufficiently large" numerically, his own student K. Borodzin proved, in 1939, that 3^14348907 is large enough. This number has 6846169 decimal digits, so checking every number under this figure would be highly infeasible with current technology. In 2002 Liu Ming-Chit and Wang Tian-Ze lowered this threshold to approximately . If every single odd number less than 10¹³⁴⁶ is shown to be the sum of three odd primes, the weak Goldbach conjecture will be effectively proved. However, the exponent is still much too large to admit checking every single number. (Computer searches have only reached as far as 10¹⁸ for the strong Goldbach conjecture, and not much further than that for the weak Goldbach conjecture) G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent English mathematician, known for his achievements in number theory and mathematical analysis. ... John Edensor Littlewood (June 9, 1885 – September 6, 1977) was a British mathematician. ... The Riemann hypothesis is one of the most important conjectures in mathematics. ... The neutrality of this article is disputed. ... Vinogradovs theorem states that any sufficiently large odd integer can be written as a sum of three prime numbers. ... In mathematics, the phrase sufficiently large is used in contexts such as: is true for sufficiently large which is actually shorthand for: there exists an such that is true for all . ...

In 1997, Deshouillers, Effinger, Te Riele and Zinoviev showed^[1] that the generalized Riemann hypothesis implies Goldbach's weak conjecture. This result combines a general statement valid for numbers greater than 10²⁰ with an extensive computer search of the small cases. The Riemann hypothesis is one of the most important conjectures in mathematics. ...

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