Goldbach's weak conjecture

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

This conjecture is called "weak" because if Goldbach's strong conjecture (concerning sums of two primes) is proven, it would be true. (Since if every even number greater than 4 is the sum of two odd primes, merely adding 3 to each even number greater than 4 will produce the odd numbers greater than 7).

In 2013, Harald Helfgott proved Goldbach's weak conjecture; previous results had already shown it to be true for all odd numbers greater than ${\displaystyle e^{3100}\approx 2\times 10^{1346}}$.

Some state the conjecture as:

This version excludes 7 = 2+2+3 because this requires the even prime 2. Helfgott's proof covers both versions of the conjecture.

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, Ivan Matveevich Vinogradov eliminated the dependency on the generalised Riemann hypothesis and proved directly (see Vinogradov's theorem) that all sufficiently large odd numbers can be expressed as the sum of three primes. Vinogradov's original proof, as it used the ineffective Siegel–Walfisz theorem, did not give a bound for "sufficiently large"; his student K. Borozdin proved that 3³¹⁵ is large enough. This number has 6,846,169 decimal digits, so checking every number under this figure would be completely infeasible.

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