In this talk I will present a work in progress related to the analysis of a Bose-Einstein condensate with three-body interactions. Roughly speaking, in an ultra cold gas of Bosons most particles share the same one-body quantum state. The accuracy of this picture depends on how much particles interact with one another. In this setting, particles see each other only if they are at least three in the same neighbourhood. We are interested in the Gross-Pitaevskii regime, where the mean-field approximation (i.i.d variables) does not hold anymore and the correlations do contribute to leading order. I will first review the usual two-body interaction case and then discuss the trivial and non-trivial extensions to the three-body case.