Problems and methods at the origin of the theory of numbers

This short report concerns some aspects of the history of number theory. These aspects have been chosen because they are strictly interconnected from a conceptual point of view. Such a connection is due to the fact that all the problems and theorems were set from a general point of view by Fermat, while before him they were treated under a methodological perspective which was neither complete nor clear. Therefore this study deals with the history of mathematical ideas and concepts and proposes a rational reconstruction of the development of some ideas in number theory. The ideal connections concerning scholars who lived in different periods are perhaps even more significant if there is no direct influence among them, because this means that some mental schemes exist independently of the undeniable influence due to circumstances. Thus some ideas on the methods used in number theory, on the logic of these methods, on the most important novelties that different approaches and methods may introduce will be proposed. The whole proposed order of ideas turns around Fermat because he was the first who framed these ideas into a complete theory. Since he clearly expressed his general perspective, but left few explicit proofs, a conspicuous part of this essay is dedicated to a modern trying to reconstruct Fermat’s methods. The guide-lines of these reconstructions will be pointed out. The present study is not only directed to specialistes in number theory or in the history of number theory, but to all the scholars who have interest in logical, methodological, gnoseological and historical-mathematical questions. Mathematical details will be explained where necessary.