Résumé : Let $S$ be a multiplicatively defined set. Ostmann conjectured, that the set of primes cannot be (nontrivially) written as a sumset $P\sim A+B$ (even in an asymptotic sense, when finitely many deviations are allowed). The author had previously proved that there is no such ternary sumset $P\sim A+B+C$ (with $\left|A\right|,\left|B\right|,\left|C\right|\ge 2$). More generally, in recent work we showed (with A. Harper) for certain multiplicatively defined sets $S$, namely those which can be treated by sieves, or those with some equidistribution condition of Bombieri-Vinogradov type, that again there is no (nontrivial) ternary decomposition $P\sim A+B+C$. As this covers the case of smooth numbers, this settles a conjecture of A.Sárközy.
Joint work with Adam J. Harper.