Youri Veniaminovich Kraskov
The Wonders of Arithmetic from Pierre Simon de Fermat
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 and for thoughts spacious.
The usual technical language does not achieve such an effect and this requires a higher level of literature accessible only to the elect, for example, such as Alexandre Dumas the Father. In one of his books Dumas even argued that writers understand history better than historians. Wherein, he has fib so famously and godlessly that historians could only smirk. However, in fact Dumas turned out be right because the lion’s share of the history set forth in the thick books did not really have place, but was simply invented and this fact also found a place in this book.
One of the features of our literary creativity is the mandatory presence in it of riddles, which are announced, but not disclosed. There is a whole 15 of such riddles in this book, and they are marked (*) in Appendix V (points 18, 19, 26, 27, 39, 41, 42, 43, 44, 45, 46, 47, 48, 69, 74). Questions and problems are focused only on key points that are of principled importance. When such a moment of truth comes, it may give the impression or even bewilderment that science has not noticed such simple things for centuries. But this is where the power of real science lies since the Almighty in his creations always follows only paths with the shortest and simplest solutions.
In real life false knowledge often takes the place of real ones. Behind this lies a lot of danger and unnecessary problems. However, they will bypass those who will be able to understand this book. However, if it will not work for someone, it’s best to turn to children for help and then they will simply be amazed at childish abilities to penetrate into corners of the unknown, which are completely inaccessible to most people. But children themselves do not realize that at their age all people are wizards for whom there are no insurmountable obstacles. Think of although a three-year-old Gauss who made accurate calculations for his father a pipeline engineer. And others can do it too, but they just don’t know about it!
In this book many different names are called, which created history by the will of Providence or case, and just because they turn special attention to themselves they deserve every respect, no matter in what circumstances and how they showed themselves because otherwise, there would simply not be events, from which the plot of our narrative was formed.
From what we have already talked about science here, it will look completely unattractive. Moreover, it will be presented as the source of all troubles, and sadly this is the harsh truth. But if the question about the place of science in society is not raised and, in any way, not clarified, then a catastrophe allegedly coming from scientists, will become inevitable and the very existence of our wonderful world will lose all meaning. This is not at all some formidable warning or apocalyptic prophecy, but merely an ascertaining that science is the only (!!!) field of human activity that predetermines all their other varieties!
Thus, in an intelligent society the highest priority of science must be ensured and supported by all available means, otherwise, it will receive only a global confrontation of ignoramuses balancing on the precarious verge of mutual destruction. And what we have now? Only that the management of society goes not in accordance with the objective laws of the world, but through blatant incompetence, irresponsibility, bribery, adventurism etc. Where is here the science? It is not even near visible anywhere. If even the applied science, which has been robbed by money-lenders to the last thread, somehow can still cling to its existence then for a fundamental science a long time ago there are no any prospects at all.
But perhaps scientists need to offer something themselves so that the fruits of their labor will be appreciated? Ha-ha-ha! There is a well-known case when Gregory Perelman had published without any conditions in free access the proof of the Poincare conjecture, which more than a hundred years apart from him could not be obtained by scientists. However, instead of (already offered to him!) a premium of US$ 1 million he got nothing. The press reported that allegedly he had himself refused under a fictitious pretext, but for some reason all thought that he was just an eccentric. However, in fact he did not even think about refusing, apparently naively believing that the prize he has fully deserved.
However, he did not take into account that in a society, in which the leading positions have not scientists, but usurers and bribe-takers, scientific discoveries that do not give immediate return with money, are even for free nobody need. In fact, they really offer prizes not for scientific discoveries, but for a well-known name that can be exploited in their own interests. Yet Perelman in this story brought the initiators of the award to a clean water after he offered to share prize with another scientist related to his scientific discovery and then it became obvious that in fact the refusal did not go at all from him, but from imaginary benefactors.
In terms of determining the value of scientific discoveries, there simply cannot be any illusions that nobody really needs them. In the obviously dying world of usury, theft, gain, speculation etc., the attitude to science can be only as it is. There is no doubt that the premium for proof of the Andrew Beal conjecture will also not be paid for its intended purpose. Are you don't believe? Well, it's very easily to check!
In this book there are examples of such calculations, which leave no room for doubts that it would be impossible to carry them out without knowing the essence … no, not of a conjecture, but of a much stronger statement called here “The Beal Theorem”! If the aim of the Beal Prize is really to get this impressive scientific discovery, then the organizing Committee in the face of "American Mathematical Society" would be easier do not rely on the propitiousness of mathematical editions, and just to request it directly from the author of this book.
This way would be clearly simpler and better since the proof of the Beal conjecture is too elementary and not so significant for science as the proof of the Beal Theorem, which would be much more useful, productive and impressive with the same end result that is required in conditions of the Beal Prize. The risk of arising another fake in this case will be excluded, but if nothing to be done to solve this problem, the initiator of the prize Mr. Andrew Beal may never wait to achieve his goal. Besides, it should be borne in mind that expert evaluation of the Beal conjecture proof does not require such obviously excessive precautions, because this task is for children from secondary school. What is written in this book is more than enough to make sure that this task has not any difficulties for the author.
It is very curious in this sense, how science will react to the appearance here of the FLT proof, performed by Fermat himself! And this is in conditions when as many as 18 (!!!) the most prestigious awards for obviously erroneous proof 1995 have already been presented! Of course, no one is immune from errors and we will show here how such pillars of science as Euclid and Gauss made the most elementary blunders in proving the Basic theorem of arithmetic, as well as Euler, who blessed the use in algebra of “complex numbers” , which are not numbers due to the fact that they do not obey to this same Basic theorem. However, Euler wasn’t aware of it yet, but his followers know this perfectly well for the two hundred years, nevertheless no one even had a finger stir to correct this mistake.
As for the not needed scientific discoveries, many people simply do not know that they can live quietly and consume all the vital resources they need only until the knowledge resource, accumulated in society for a given level of its development, will be exhausted. And after that, in order to keep what has been achieved, the stronger countries will attack the weaker ones and live at the expense of their plunder. But this would not have been necessary at all if these “strong” countries had enough knowledge. Then they would not have conflict with the rest of the world since all the necessary resources would be provided in abundance by science.
On this we will complete our introduction, but we will give it such a secret impulse that will allow us to perform a real wonder! … no, even two! We can call these wonders here by their proper names because our eternal opponents from the complete lack of real science by them, are simply incapable of this.
As a result, they will learn about the realization of the most grandiose technological breakthrough in Russia in the entire history of our civilization, with unlimited potential of development effectiveness for the immense future. The notorious “valleys”, “techno parks”, “incubators” and the like ghosts for such breakthroughs are unsuitable in principle. But still earlier, another wonder will happen when Russia literally in a couple of months, on the wreckage of collapsing today the world usury financial system, will create a new one, in which no any international money will be needed and all countries in international trade will use only their national currencies.
Are you again don't believe? Well, you can see for yourself because the book is in your hands!