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 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.) However, this bound is small enough that any single odd number below the bound can be verified by existing primality tests such as elliptic curve primality proving, which generates a proof of primality and has been used on numbers with as many as 20,562 digits.

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.

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