Résumé
Our thesis deals with the transposition of mathematician's reserach activity in mathematical classroom, in the domain of number theory. Our research focuses on the study of a research process for researchers, pupils and students involved in the research of an unsolved problem: the Erdös-Straus conjecture. Our mathematical and epistemological analyses allow us to identify different aspects of the mathematician's work and the elements for progress in his research. The notion of "gesture" is developed to describe, analyze and contextualize different research processes. This analysis reveals the potentiality of this problem to create a research situation in classroom, where pupils are in a position similar to the mathematician's one. Didactical analyses are based on the construction of such a situation and its experimentation in laboratory. We study the research process of the students with the methodological tools developed in mathematical and epistemological analyses. This analysis shows several potentiality of this situation: a wealth of procedures implemented, effective work on the dialectical aspects of the mathematical research activity and implementation of experimental approach. The notion of "gesture" is relevant to consider the question of the transposition of mathematician's work.