Goldbach's weak conjecture

In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture or the 3-primes problem, states that:

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

or equivalently:

• Every odd number greater than 5 can be expressed as the sum of three 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 >4 is the sum of two odd primes, merely adding three to each even number >4 will produce the odd numbers >7.)

The conjecture has not yet been proved, but there have been some helpful near misses. In 1923, Hardy and Littlewood showed that, assuming a certain generalization of the Riemann hypothesis, the odd Goldbach conjecture is true for all sufficiently large odd numbers. In 1937, a Russian mathematician, Ivan Vinogradov[?], was able to eliminate the dependency on the Riemann hypothesis and proved directly that all sufficiently large odd numbers can be expressed as the sum of three primes. Although Vinogradov was unable to say what "sufficiently large" actually meant, 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. Fortunately, in 1989 Wang and Chen lowered this upper bound to 10^43,000. If every single odd number less than 10^43,000 is shown to be the sum of three odd primes, the weak Goldbach conjecture is effectively proved! However, the exponent still needs to be reduced a good deal before it is possible to simply check every single number.

In 1997, Deshouillers, Effinger, Te Riele and Zinoviev showed 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.

External links and references

• 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