Youri Veniaminovich Kraskov
The Wonders of Arithmetic from Pierre Simon de Fermat
Abstract
Within the framework of the designated topic:
1. There are restored some little-known facts from the Pierre Fermat's biography.
2. For the first time, there are given two definitions (mathematical and general) the concept of number, as well as a new versions axioms and basic properties of numbers following from them.
3. There is shown fallacy of Euclid, Gauss, and Zermelo's proofs of the Basic Theorem of Arithmetic (BTA), without which the foundation of the whole science collapses.
4. There are given comments on Zermelo's proof of the Basic Theorem of Arithmetic.
5. There is restored the Fermat’s BTA proof by the descent method.
6. There is restored a simplest method of proving the FLT for the 4th power.
7. There is shown the fallacy of the G. Frey’s idea lied in the basis of the A. Wiles’ FLT proof 1995.
8. There is restored the Fermat’s proof of his Last Theorem on the basis of a new way to solving the Pythagoras’ equation x2+y2=z2 by using the formula, discovered by him and called Fermat Binomial.
9. As a consequence of the FLT proof there are formulated Theorems on Magic Numbers, the validity of which is confirmed by examples of calculations.
10. There is proposed the formulation of the Beal Theorem revealing the essence of the Beal Conjecture for equation Ax+By=Cz. There are given examples of calculations according to this theorem.
11. There is restored a way for proving the Fermat's Golden Theorem.
12. There is restored the proof’s method of the Fermat's grandiose discovery about primes of the form 4n+1.
13. There is restored a way of solving the Archimedes-Fermat equation Ax2+1=y2.
14. There is restored a proof of the Fermat's theorem on the unique solution of the equation y3=x2+2
15. There are shown examples to application of methods for solving problems proposed by Fermat.
16. There is shown a role of arithmetic as the basis of foundations of the whole science.
17. There are shown examples of non-existent sciences such as history, informatics and economics.
18. There are given definitions the essence of the basic concepts’ informatics and economics.
19. For the first time, there is given a general definition the essence of the concept an information.
20. There are proposed some fundamentals of informatics as a science.
21. There is proposed a method for ordering knowledge using the Basic law of systems.
22. There is proposed the idea of an economic breakthrough based on the new generation of IT.
23. There is proposed the new essential understanding of money and their functions.
24. There is proposed an International Payment System (IPS) of a fundamentally new type using national currencies in international settlements.
25. There is given essential understanding the source of profit on invested capital.
26. Historical episodes are presented in a stylized literary form.
27. There is proposed the restoration of the tombstone of P. Fermat with an English translation.
28. There are proposed 15 riddles in Fermat-style i.e. without their complete solution.
29. There is compiled a full list of scientific problems presented in this book in 100 points.
30. The original Russian text of the book is translated by its author into English.
The book is intended for a wide range of readers.
Feedback and comments may be sent to c_city2000@mail.ru
From the Author
Imagine that you decided to write a book and such an interesting one so that, wow, it is breathtaking! But how to begin? It's very simple, you open this book and immediately see the famous picture with the portrait of our main hero the legendary Pierre de Fermat from the complete collection of all his written works, that Paul Tannery and Charles Henry released at the beginning of the 20th century. So here this picture is now decorated with a wonderful equation called Fermat’s binomial formula, about which the current science still has no idea, although this formula was published back in 2008.
Pic. 1. Portrait of the Senator Pierre de Fermat
And the next picture is a delightful sculptural composition. Our hero is so good-looking and next to him is his muse, from her he drew his inspiration and created such wonders, from which the entire civilized world since the XVII century and until now just goes crazy.
Pic. 2. Pierre Fermat and His Muse in the Capitol of Toulouse
After this picture follows the image of a page from Diophantus' “Arithmetic” with the text of the Fermat’s Last Theorem, published in 1670, and from here immediately the main theme appears.
Pic. 3. Diophantus’s “Arithmetic” Page of the Greek-Latin Edition 1670 with the Task VIII and a Remark to it, Becoming Later the Fermat’s Last Theorem
But this is about science and can it really be something interesting?
In the one, that we are taught, it is hardly, but in the true science there can be really amazing wonders! Because, unlike conventional belles-lettres, science is not just a literary embodiment of the author’s idea, everything here is much more complicated, since he should reflect not invented by him, but the truest reality, which can always be checked and if something is wrong, then the whole work will go down the drain. Such are the cruel laws of this genre. However, really, if you just look at the content of the book … there are some problems, tasks, refutations … outwardly it looks somehow not very intriguing. But the impression will change when we after the two wonderful pictures at once take the bull by the horns – we give another picture by our hero with his really written marks in the margins of one old book …
Pic. 4. Fermat’s Recording in Margins of Apollonius the Perganos'
Book, named Conic Section
and right behind it suddenly … bubuh!!!
Pic. 5. The Page of the Diophantus’ “Arithmetic” Edition 1621 and the Restored Text of the Fermat Last Theorem in the Margins
Oh, what is it??? … This is just not possible!!! Even eyes cannot believe. A page from Diophantus’ “Arithmetic” only an earlier edition 1621 from Fermat’s personal copy of the book with the most famous theorem on the margins, from which the entire scientific world is still in a fever! Everyone knows this book has been lost! But where does it come from?
From there! The book was so well hidden that so far no one knew about it!
Yeah … and what's next?
The next is that Fermat’s binomial formula is from this book of ours, where it will be derived, and it is easily to make sure that formula is true. Take any two numbers x, y and the third z their sum (or difference), substitute in the formula and everything will be OK! Isn't that curious?
Well, but what is the avail of this formula and what is its significance?
But without it the Fermat Last Theorem will remain unproven!
Yeah… would Fermat really hide such a treasure from everyone?
Well, that's a good question. Anyway, the plot is already twisted in the right direction and quite famously … although this is not even the beginning and only a light aperitif, but it acting already is oh so good!
Is it true that a lot of money was promised for this theorem?
Hmm, promised – yes, and even paid … but not at all for the FLT!
How is this so, and for what?
For the error, that was revealed in the proof, but science did not want notice it for 25 years.
So, it turns out all this work with our book will go past the cash box?
Well, that's another good question. However, it has already long been known that for science money is not paid. Well, if … as if for … doing science … or taking care of it … well, for the good of the state, then yes, billions are spinning here, but the very science has never existed and does not exist now.
So perhaps we need to do something incredible here in this book so that we can get something out of these billions?
But no, then no one will read this book, besides all the places of the distribution of money have long been occupied. It is necessary to do this book with real science, but for this we need to use special approaches and principles. Particularly, the main principle should be that all the topics touched upon here contain secrets, and such that they are hidden not otherwise … as it is said … ah yes, behind seven seals!
Wow, it’s light to say…
Moreover, about this should be announced in advance (we are now doing this) that even in relatively calm places of narration no one should have any doubt that in every topic something must necessarily be very much surprise or even shake! Well, not literally, of course. Say, it is given in some place a calm explanation of some entity or their components and suddenly, unexpectedly … bubuh!!! And it turns out that here the amazing secret is hiding, which was lurking under the guise of what has been known for a long time, but did not attract any attention. Another principle is that of all the secrets we have revealed (and there are more than a hundred of them here!!!) we must leave some share (say, one and half dozen) as undisclosed. Otherwise the children will be offended that them no one nothing left for their own discoveries …
!!!… What?… The children???
Oh yes, we narrowly forgot to say that this book is largely childish. Most adults would be unable to retrain for a such a thorough science. It is clear, they will very offended hearing this, but we of course will not trumpet about it over the entire world, and even will observe decency i.e. from all the tasks of basic science that we will discuss, only the lightest will be singled out specifically for children. They’ll be happy about this because among these tasks there will be … the very FLT!!! Yes, it is in this book of ours that for the first time in the entire history of the FLT, this problem, and not just a problem, but one of the fundamental basics of science (!!!) will be solved in expanded form together with some other problems related to it. In this sense, the FLT will be our guiding star and its first-discoverer Pierre Fermat will be our mentor from whom we will directly (!) receive all the information we need.
However, the FLT is only one of a whole hundred (!!!) tasks that will also receive solutions here for the first time! Along with the solution of the FLT problem, there will be a whole heap of other very impressive tasks that will delight many kids and interest them for a long time. And we will try very hard to get their attention, then they also will draw adults into this game. If we compare the significance of the tasks solved here, the undisputed leadership will remain by the FLT although as for the difficulty of its solving, it is quite simple and secondary school students can easily cope with it … Do you doubt? But still this is a task from the 17th century, however today's ones will be more difficult, but in order to solve them, you need to have a reliable foundation of knowledge and FLT is one of its cornerstones.
About other tasks we will else tell in detail and they will then cease to be a secret behind seven seals, but those that we only call and will not reveal to the end, will be very curious. Take for example one problem of astronomers who have accumulated huge collections with records of cosmic signals of electromagnetic radiation from all points of the starry sky. And what to do with them nobody knows. Whether it is the brothers in mind communicate with us or it is just natural radiation. Unmeasured money spent on giant radio telescopes and a bunch of people working for them, but there’s nothing to use. They are simple the poor fellows!
But we know a way how to distinguish reasonable signals from natural ones, but we will lowly keep silent about this. We know this only because we will reveal here one of the greatest science’s secrets about determining the essence of the information’s phenomenon. It is clear the current science in this matter doesn’t know anything at all. But as soon as we clarify it, then besides us also others can then guess how electromagnetic signals may be analyzed for determining their reasonable origin.
We'll begin to approach the solution of this problem by penetrating the most important problem of all science, this is defining the essence of the number phenomenon, about which unfortunately science also does not know anything like about many other “simple” questions. And now, when it comes to the most important essence capable of displaying all other essences (yes, yes, it's about number!), we can already see that case is taking a much more serious turn and such shifts are awaiting us, which were not even in the entire visible history of our civilization!!!
Again, are you in doubt? But that is why you would never guess what are those shifts, and when you find out, you can hardly believe that they are possible, but secrets behind seven seals cannot be other. Until someone gets to them and begins to slowly reveal them, everyone will remain in the dark about how a much is arranged in this world and why it is better to know about it than to live in ignorance about the existence of the wonderful world of science where perfection and precision dominate. And it is just curious to find out how, for example, does look the proofs of the Fermat’s theorems that turned out to be inaccessible even to the greatest scientists. Now there is a book that allows everyone to see with one's own eyes all these hidden and inmost secrets unknown to current science and thereby receive a unique opportunity to rise to a height unattainable before.
It is enough to see the name Fermat in the title of our work to suppose something great in it. It was such a remarkable man that he could not create anything petty, even average: his mind shone with such a brilliance that he could not tolerate anything dark. It may be said that he is similar to the sun in a moment driving off the dusk and spilling the blinding light of its bright rays even into the abysm. Until now, everyone has been amazed by Diophantus and this is well deserved; but, no matter how great he was, it is a pygmy in comparison with such a giant who has come a long way around the world of mathematics traversed lands that have never been seen before. Vieta was praised by all those who in our century devoted themselves to the study of algebraic operations, so for the glorification of some scientist it was enough to say that in the work on analysis he followed the thoughts of this author. But he also did not reach the heights of science which will become clear from the many examples explained below. Before Claude Gaspar Bachet I always bowed down as before a man of the subtlest mind; in addition, he was a close friend of mine and his research on Diophantus perfectly shows how astute he was in the science of numbers. But his gaze is weaker if you compare it with the lynx eyes of our Fermat, which penetrated into the most intimate depths.
Jacques de Billy, 1670
Priest and Professor of Mathematics
Introduction
In the content of the book is presented the main theme consisting of about three tens items. This would be nothing special if all these items did not contain … the most real and incredibly loud sensations! But to say only this about this book would be to say nothing about it. Alone only illustration of the real (!) text in the margins of a missing book (see Pic. 5) we have restored, can cause a real shock among experts of the main theme! They might think: "Is this really the same book with Pierre Fermat's notes in the margins?". But no, this book is not yet available. And since we still managed to find out, what was actually written in its margins where Fermat's Last Theorem should be located, we depicted this recording by all means available to us. If we compare this restored text with the one that was published back in 1670 (see Pic. 3), then it becomes obvious that these are completely different recordings!
However, in our time, the Internet is also literally flooded with heart-rending screaming headlines about some sensations, which in fact are not, and their distributors resort to them only to raise the statistics of browsing. When it comes to science, if there are really sensations, then only in doses that cannot be captured by any statistics. The problem here is that the evaluations in the headlines are given by the distributors of information themselves who obviously should not be trusted. As for the content of this book, the situation here is principally different, since all the data here, assessments and conclusions can be checked by the most objective and incorruptible judge i.e. a regular calculator and anyone can always refer to it.
In particular, if there is a suspicion that the restored Fermat’s record on the margins is nothing more than another fake among the sea of any other ones, they will prove to be not only nonconstructive, but also rejecting the opportunity itself to find out the real solution of the famous scientific problem. If this factor is not taken into account, then those who persist in such suspicions risk being in a very stupid position, since in this restored recording there is exactly what science still had no idea about. In fact, for science the FLT has always been just a puzzle, which for more than three centuries, could not be solved.
Such a scornful attribution of one of the fundamental scientific problems to the sphere of intellectual entertainment led to the fact that real science began to give way to ideas that have nothing to do with it. As a result, it turned out that all reference books and encyclopedias in unison and categorically tell us that the FLT problem has long been solved, but in fact science has no idea about how things really are. If this were indeed the case, the consequences would be so significant that they would radically change the state of all science in general as a whole!!!
Are you not believe? Well, judge for yourself, here is just one of these consequences. If the FLT is proven i.e. the solution in integers of the Fermat equation an+bn=cn for n>2 is impossible, this equation turns out to be the only (!!!) exception from the more general case Ax+By=Cz in which for any (!!!) given natural numbers x, y, z except of course x=y=z>2 may be calculate any number (!!!) of solutions in integers! And what now? Does science know, how to solve this general equation? Of course, no. Or perhaps science at least knows something about Fermat’s equations for children with magic numbers? Or about the wonderful Fermat’s binomial formula? Also no. However, the Soviet science fiction writer Alexander Kazantsev somehow incredibly way guessed about this formula, but mathematicians could not help him to derive it, so instead of a spectacular equation (see Pic. 1), he had to demonstrate an empty dummy.
Apparently, he did not even suspect that he had to ask for help not from mathematicians, but from children, then the result of his fantastic guesstimate would have appeared much earlier than this book where this formula is derived exactly in the appropriate place i.e. in the restored FLT proof from the Fermat itself! If this proof (obtained 365 years ago!!!), will learned by children studying in ordinary secondary school, they can easily cope with solutions of equations containing the magic numbers. These numbers, unlike some that mathematicians work with, are real because they obey to the Basic theorem of arithmetic (BTA). But the trouble is that current science does not even suspect that this most fundamental of all theorems has not been proven up to now!!!
But if science had become aware of this, then it would have no other choice as to accept BTA as an axiom since otherwise, science itself would simply disappear and then it could not be at all! Now, it will be a real surprise for science to find out that the problem of BTA proof was solved by the same Pierre Fermat and for this he used his own brand called the “descent method”. However, he could not divulge his proof since this would indicate an error of Euclid, in the proof of which he had it noticed, but this, not only at that time, as well as even now is inadmissible since gods by definition cannot be mistaken. It is also curious that without noticing the presence of BTA in the Euclid’s “Elements”, even such a giant of science as Karl Gauss exactly repeated the error of Euclid, what apparently also indicates his true divine origin.
In this book the proof of BTA obtained by Fermat is now like the FLT restored and the loopholes for penetrating into science of all sorts of pseudo numbers are closed, although it will not be easily to cleanse them because the precedent for them was created by none other than the greatest scientist and mathematician Leonard Euler! Indirectly in this was also involved Karl Gauss proving the “basic theorem of algebra”, which without these allegedly numbers called “imaginary” or “complex” would be wrong. Long before Euler and Gauss such well-known scientists as Leibniz and Cardano expressed their categorical rejection to this kind of "numbers". But they did not know that these Kazantsev’s non-existent beings disobey to BTA since only in 1847 Ernst Kummer told this very unpleasant news for the first time to the entire scientific world. However, for some reason this scientific world up to now stubbornly unwilling to get rid of the illusion of what really doesn't exist at all! For example, the Euler’s formula that causes delight eiπ+1=0 is in fact a complete nonsense that has nothing to do with science except perhaps to teach children not to believe in the reality of such tricks. Here even to them it is obvious that eiπ = -1 and this is certainly an obvious bullshit since the imaginary number i = √-1 being here makes imaginary and meaningless everything in where it is presented.
The main hero of our narration Pierre Fermat even in terrible dreams could not have imagined that only one of a whole hundred of his tasks [30] could even 325 years after the first publication of his works so much to discredit science, that it will turn out not only be incapacitated, but also literally standing in an head over heels position!!! Just in the period 1993-1995 it occurred immediately two events related to the FLT. The first is the Andrew Beal conjecture about the equation Ax+By=Cz, the proof of which allegedly allows to get FLT proof in one sentence. And the second is the Andrew Wiles’ FLT “proof” (which up to now nobody had understood), the news of which appeared in some incredible way in the newspaper "The New York Times" two years ahead of it! But then it was simply impossible to imagine what would happen when 25 years later it was suddenly found out that both of these events are pure misunderstandings!!!
Beal conjecture to the difficulty of its proof is suitable perhaps for school-age children. But this is just incomprehensible to the mind how it could not be proven up to now even for a prize of a whole million dollars!!! Another no less surprising side of this conjecture is the lack of under-standing of how it is related to the proof of FLT, since what is written on this subject in Wikipedia is completely absurd. Nevertheless, Andrew Beal establishing such a large premium for his conjecture, clearly deserves universal respect, since with such a step he drew the attention of science on a theme, which had already taken place at Fermat in the above-mentioned restored FLT recording on Pic. 5.
The announced competition to prove the Beal conjecture does not allow us to clarify the solution of this problem in this book, because it can cause a real stir in the scientific world. Despite the simplicity of the proof of this conjecture, its consequences will be a loud sensation, since they will allow us really to get the simplest proof of the FLT. On the other hand, this will be too modest a result for the Beal conjecture, because its scientific potential is incomparably more powerful and impressive. To fix this situation to the best, this book will offer a more meaningful formulation of this problem, which called here the Beal Theorem, that not only confirms the correctness of conjecture, but also opens up the possibility of solving the equation Ax+By=Cz for any natural powers except the case x=y=z>2.
As for the Wiles’ FLT “proof”, it rests only on the Gerhard Frey’s idea, where again (for the umpteenth time 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.