Goldbach's weak conjecture

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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:

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.)

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.)

The conjecture has not yet been proved, 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 that 3^14,348,907 is an upper bound for the threshold that determines if a number is large. This number has more than six million digits, so checking every number under this figure would be impossible. In 2002 Liu Ming-Chit and Wang Tian-Ze lowered this upper bound to approximately 10¹³⁴⁶. 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.

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.

[edit] External links and references

1. ^ Deshouillers, Effinger, Te Riele and Zinoviev, "A complete Vinogradov 3-primes theorem under the Riemann hypothesis", Electronic Research Announcements of the American Mathematical Society, Vol 3, pp. 99-104 (1997). Available online at http://www.ams.org/era/1997-03-15/S1079-6762-97-00031-0/S1079-6762-97-00031-0.pdf