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.
