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in the past 350 years!) an elementary error was made!!! In this case, if something has been proven it is the complete inability of science to notice such errors, which must be teaching by schoolchildren. As a result, these events took place in such a way that on the FLT problem and its generalization in the form of the Beal conjecture, science once again became a victim of misunderstandings i.e. the current situation with the solution of the FLT problem is no better than the one that was 170 years ago, when the German mathematician Ernst Kummer provided proof of the FLT particular cases for prime numbers from the first hundred of the natural numbers.

      With a such amount of knowledge available to current science, its helpless state seems as something irrational and even unthinkable. Nevertheless, it permeates whole of it through and far from only the FLT problem, but also in general wherever you poke, the same thing happens everywhere – science shows its inconsistency so often and in so many questions that they simply cannot be counted. The only difference is that some of them still find their solution, but with the FLT science has been stuck for centuries. However, the greatness of this problem lies in the fact that it, apart from purely methodological difficulties, points to some aspects of a fundamental nature, which have such a powerful potential that, if it succeeds in uncovering of it, science will be able to make an unprecedented breakthrough in its development.

      Fermat paid attention to this aspect and was the first to notice even then, that science had no roots to support it as a whole. Simply put, the logical constructions used in solving specific problems do not have a solid support that determines the way, in which each separate branch of knowledge exists. If there is no such support, then science has no protection from the appearance of all kinds of ghosts taken as real entities. The Basic or as it is also called Fundamental Theorem of arithmetic is a vivid for it example. It would seem, what is simpler, one needs only to accept as an unchangeable rule that the numbers can be either natural ones or derived from them. Anything that does not obey this rule cannot be a number. Given that arithmetic is the only science that no other science can do without, it can be stated that all science cannot do without BTA at all! But science itself is not even aware of the fact that BTA is still not proven. And how do you think why? … This is because science simply does not know what is a number!!!

      Even to people far from science, this obvious fact can make a shocking impression. Then the question obviously arises: if science does not know even this, then what can it generally know? In this book we’ll explain what the difficulty is here and suggest a solution to this problem. This immediately draws the need for axioms and basic properties of numbers, which were also previously known, but in a very different understanding. After the definition the notion of number and axiomatics, proof of the BTA is required, since otherwise, most of the other theorems simply could not be proven.

      As can be seen from this example, if a fundamental definition the concept of a number is given, then immediately a need appears to build an initial system defining the boundaries of knowledge, in which it can develop. It’s like by musicians, if there is an initial melody, then the composer can create a complete work of any form and type from it, but if there is no such melody then there cannot be any music at all. In this sense, science is a very large lot of different melodies piled up into a one bunch, in which science itself is completely entangled and stuck.

      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

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