  In this paper we develop the interpolating cavity field technique for the
mean field ferromagnetic p-spin. The model we introduce is a natural extension
of the diluted Curie-Weiss model to p>2 spin interactions. Several properties
of the free energy are analyzed and, in particular, we show that it recovers
the expressions already known for p=2 models and for p>2 fully connected
models. Further, as the model lacks criticality, we present extensive numerical
simulations to evidence the presence of a first order phase transition and
deepen the behavior at the transition line. Overall, a good agreement is
obtained among analytical results, numerics and previous works.
