# Goldbach's Weak Conjecture

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## Theorem

Every odd integer greater than $7$ is the sum of three odd primes.

## Proof

## Also known as

It is also known as the **odd Goldbach conjecture**, the **ternary Goldbach problem**, or the **3-primes problem**

## Historical Note

- 1923: It was proved by Godfrey Harold Hardy and John Edensor Littlewood that the Generalized Riemann Hypothesis implies Goldbach's Weak Conjecture for sufficiently large numbers.

- 1937: It was proved by Ivan Matveevich Vinogradov, independently of the Generalized Riemann Hypothesis, that all sufficiently large numbers can be expressed as the sum of three primes.

- 1939: Vinogradov's student Konstantin Borodzin proved that $3^{14348907}$ is large enough.

- 1997: Proved by Jean-Marc Deshouillers, Gove Wilkins Effinger, Hermanus te Riele and Dima Zinoviev that the Generalized Riemann Hypothesis implies Goldbach's Weak Conjecture.

- 2002: Liu Ming-Chit and Wang Tian-Ze lowered the threshold for Vinogradov's result to approximately $n > e^{3100}$, which is roughly $2 \times 10^{1346}$. This gives an upper bound such that it is feasible to test any single odd number below that threshold. Though one can calculate: if a modern desktop were the size of a 1cm cube, one filled the solar system with such cubes and left them running the fastest known algorithms since the beginning of time, numbers larger than $10^{70}$ would remain untested. For this reason the problem is still considered unsolved, though the existence of only finitely many counterexamples is a worthwhile result in its own right.

- 2013: Harald Andrés Helfgott proves the conjecture.

## Also see

- Goldbach Conjecture
- Goldbach's Marginal Conjecture
- Goldbach Conjecture implies Weak Goldbach Conjecture, which is why this one is called weak.

## Source of Name

This entry was named for Christian Goldbach.