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Abstract
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Since at least twenty five centuries, prime numbers have been a fascinating subject, where conjectures are easy to formulate and theorems difficult to prove. Last Spring,two outstanding questions have been solved. First, Zhang Yitang proved that the difference between consecutive primes does not tend to infinity. Second, Harald Helfgott proved that any odd integer larger than 5 can be expressed as a sum of three primes, a question explicitly formulated by Goldbach as soon as 1742. As I understand, Zhang Yitang gave a series of lectures in Beijing on his remarkable result. I shall thus focus on Helfgott's theorem, presenting the history of the problem, its connection with Zhang's theorem, the ways for approaching it, the past results and giving some hints on the new ingredients introduced by Harald Helgott.
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