In order to show how effective the descent method may be, in this book in addition to the proof of BTA, it was also restored proof by the same Fermat's method a theorem about the only solution of the equation y^>3 = x^>2 + 2 in integers, which could not be proven until the end XX century when André Weil has make it, but by another method and again of the same Fermat. If the problem proposed to Wallis had also been solved by descent method then the three greatest mathematicians, close to the Royal courts, would not have to work so hard. However, the result that they were able to achieve, may sink into oblivion due to excessive difficulties in understanding it and then all this gigantic work will slowly bypass the manuals as had already happened with the Cauchy proof of the Fermat’s Golden theorem, about which it will also be told here.

There will also be touched upon a theme, which because of its seeming extreme difficulty, was as if ones did not notice and evade it. This theme about the special significance of arithmetic for the formation an abstract thinking, which obviously is of exceptional importance not only from the point of view of studying in the field of education, but also for understanding the essence of such a notion as mind. Having no such understanding, science as well as the story with imaginary numbers, is doomed to many failures. In particular, all attempts to create "artificial intelligence" of non-biological type will be in vain since it is impossible in principle! It will be shown in this book how Gottfried Leibnitz’s truly ingenious conjecture, that thinking is an unconscious process of calculations, turned out to be true although only somewhat, because the mind cannot exist as a separate object or device and is a phenomenon of an ecumenical scale!!! If we now try to resume everything that we have mentioned here regarding arithmetic, then it will become clear, this is not only a science of sciences, but also a very effective sample for imitation.

Of course, in its present state it would be simply unthinkable, but taking into account what is stated in this book, such an imitation will become inevitable and a certain standard will be created, by which all sciences without exception will be built. It is not difficult to guess that the first point of this standard will be the definition the essence of given specific science. And of course, everyone will immediately think that it’s very easily to find an answer to such a question at least by looking in some reference books or encyclopedias.

Aha, if it were so! Not to mention that the answers to this simple question for some reason turn out to be different (?), and to understand at least something from all them is hardly possible. Then it turns out that scientists specializing in some sciences simply do not know what they are doing? No, of course. They also like their predecessors use terminology, the meaning of which for some reason no one bothered to define and as a result of such a game without rules, sooner or later ghosts arise, which create the illusion of fantastic progress.

Well, and what about the sample for imitation? Considering the fact that in this book there is not even one, but whole two definitions of the essence the notion of a number, it is possible on this basis to formulate a brief definition the essence of arithmetic, say so: arithmetic is the science about the origin of numbers and methods of computations. Then from understanding the essence of numbers, one can construct their axiomatics and basic properties, which in turn will lead to BTA and other theorems arising from the needs for computations. In a similar way you can build also other knowledge beginning with basic notions and an essence of the science built on them.