But if science is built within the framework of the system laid down in it initially, then it will be as an unaffordable luxury a situation, when each individual task will be solved only by one method found specifically for it. The same problem took place in the days of Fermat, but for some reason besides him no one then bothered with it. Perhaps therefore, the tasks that he proposed looked so difficult, that it was not clear not only how to solve them, but even from which side to approach to them.

Take for example only one of Fermat’s tasks, at the solution of which the great English mathematician John Wallis turned out properly to calculate the required numbers and even get praise from Fermat himself, any his task in that time nobody could solve. However, Wallis could not prove that the Euclidean method, applied by him, will be sufficient in all cases. A whole century later, Leonard Euler took up this problem, but he was also unable to bring it to the end. And only the next royal mathematician Joseph Lagrange had finally received the required proof. Even after all these titanic efforts of the great royal trinity, for some reason it remained unattended Fermat's letter, where he reported that the task is solved without any problems by the descent method, but how, nobody knows up to now!

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 y³ = x² + 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.

Now for example, we need to use arithmetic as a sample for imitation in order to build, say, physics. To do this, we take as one of the basic definitions to this science as follows: Physics is the science about the essence, properties and interaction of material objects. Hmm … It seems here we stumbled upon an insurmountable wall because the definition the notion of matter does not exist. Philosophers spent a huge lot of paper, but all this without some use. However, as popular wisdom says, there is nothing to blame on others if they themselves have curved mugs. Physicists themselves can solve this problem without any special difficulties because no one else will do it for them.

They simply accept as an axiom that all consisting from matter has such properties as mass and energy and so simple the whole problem will be solved. Well, and what about the definition the essence of these properties themselves? But this is still Sir Isaac Newton has very well worked and even used the style of presentation along with approaches from Euclid himself! And now, standing on their shoulders, it’s not at all difficult for us to reveal the essence of these notions especially after physicists have the problem with the units of measurement solved. Indeed, in arithmetic it is only implied that all calculations must be carried out in the appropriate units of measurement while in other sciences these units must always be concrete.

For example, in informatics Bit is used as the unit of measure, but here scientists also screwed up. Since the times of Claude Shannon, it is considered that the quantity of information is measured by Bits, but given that the notion of information is not defined at all, it turns out that they measure unknown what. However, in fact, it is all very well known to everyone that by Bits is measured the capacity of memory of information carrier. But how to measure the quantity of information itself is a problem, the solution of which will largely determine the possibility of implementing the most powerful technological breakthrough in the entire history of our civilization!!!

A term "technological breakthrough" is from the field of economics, but this science is only a ghost if only because it uses as units of measurement only meaningless titles. Economic crises in contrast to the devastating storms, hurricanes and tornadoes, have no natural origin since they are the consequences of people activity who do not understand what they are doing and therefore are not able to prevent them. This book will offer a way to solve these problems from the point of view of the possibilities of building not sham ones, as they are now, but real informatics and economics built in the image and likeness of arithmetic.

From that we have already said, many people will probably think that all this looks like something too fantastic to be a reality. But everyone so thought also about Fermat. When he offered his task to someone, that someone discourses very simply: well, if Fermat is a Gascon, it means a prankster. In Simon Singh’s book about the FLT, Descartes allegedly called Fermat a boast man, what confirms this common opinion, but his exact phrase was: “… unlike Monsieur Fermat, I’m not a Gascon”. If this introduction of ours also will causes distrust or will be perceived as humor then this is exactly what is needed, because it consistent with the spirit of our main hero.

On the other hand, all the themes touched here, are too fundamental to be disclosed in the traditional style of scientific monographs. Then it would have turned out something like, say, the British Encyclopedia or the complete works of Leonard Euler consisting of about 800 volumes, which for more than 200 years anyone had not been able to publish completely at least once. So that our works would not be lost at all, we had to take an extraordinary step i.e. to use for this book an unusual literary genre called here the scientific blockbuster – a combination of narrative in the sharp style of artistic prose along with the separate fragments of purely scientific content.

How one would not to relate to this kind of innovation, here is the result already evident: main themes of the book’s content are presented in detail in 6 points of the “Resume” section and 100 points in the list of Appendix V, which is made up of what will be clearly new to current science. In addition, in order to densify the main content, 172 comments were carried out and three separate miniatures were added as applications, which usually have a reference character but here, they are presented as a natural continuation of the main part of the book, without which it would be incomplete.

The plot of the first miniature is very interesting because in the proof of BTA from German professor Ernst Zermelo (a student of Max Planck himself!) 1912 there is such a barely noticeable error that upon learning of this the authors of the textbooks will be extremely surprised. But no less surprising here is the fact that this error in fact is the same as in the Gerhard Frey’s idea for the FLT "proof" by Andrew Wiles 1995 only more veiled. Thus, the mistake coming from 1912 and appear in 1993 turned out with just terrible consequences, which completely destroyed the "solutions" of two fundamental problems that the scientific world so carelessly allowed himself to admit.

The second miniature is no less curious, because it describes in detail two proofs of the same particular case of FLT for n=4, first by Leonard Euler and then by Pierre Fermat in the reconstruction of I. Bashmakova. Both proof as twin brothers are built on the Pythagoreans identity and in both the descent method is used. They differ only in the intricacies the logic of output to the same end result. These intricacies, although different, are quite complex, what indicates the highest skill of their authors. But the end of this miniature is simply amazing. And indeed, this proof can be obtained from the same Pythagoreans identity literally in a one line (!!!), and this very line is just in the FLT recording we restored in the margins of the book shown in Pic. 5.

In addition to the Euler proof of the special case of FLT presented here, it is also added the full text of all Euler's proofs related to Fermat's grandiose discovery of the truly magnificent properties of primes 4n+1 type. This work required the utmost exertion of all Euler's creative and physical powers within seven years, but the most important proof that these numbers always consist of the sum of two unique squares, is presented by him in such a way that it is unlikely that anyone except himself understands its essence. From Euler's letter to Goldbach with this proof, at first no one understood anything at all, and after the corrected version received by Goldbach in another letter, all the experts tacitly accepted his proof, although it is far from obvious and besides, numbers of this type should be the sum of two unique squares, but about it there is not a single word at all.

Finally, the third miniature is a journey into the past. There will be a lot of surprising and even shocking things, but here we will pay attention to only one a moment. This is the Fermat’s proof of his most grandiose discovery in the field of prime numbers, which is unknown until now and here it is presented by a special way and in amazingly beautiful form. The story about this through the mouth of Fermat's son Clement Samuel with a cherry on the cake in the form of a spectacular equation will make just as colorful an impression as the beauty of nature.

The method of vestment in the verbal form of the content of this book, chosen by us, although it requires an immense strain of all forces from the author, still yields a result, in which a small volume of the book carries the knowledge of thousands of scientific monographs! Perhaps such a precedent will be the first and the last, and in this sense, it is not a competitor to traditional scientific monographs. However, in essence it is just following the simple advice of a classic on choosing a style of exposition, where for words it would be cramped