The Wonders of Arithmetic from Pierre Simon de Fermat

Youri Veniaminovich Kraskov (2021)

This book shows how the famous scientific problem called "Fermat Last theorem" (FLT) allows us to reveal the insolvency and incapacity of science, in which arithmetic for various historical reasons has lost the status of the primary basis of all knowledge. The unusual genre of the book was called "Scientific Blockbuster", what means a combination of an action-packed narrative in the style of fiction with individual fragments of purely scientific content. The original Russian text of this book is translated into English by its author Youri Kraskov.

2. The History of Delusions

An unprecedented succession of failures, wrecks of secret hopes and defeats in the protracted for centuries storming of an impregnable fortress under name the Fermat's Last Theorem, turned into a such nightmare for science that even its very existence have been questioned. Like the fierce plague epidemic, the FLT not only deprived the minds of numerous amateur fermatists, scientists and unrecognized geniuses, but also very much contributed to the fact that the whole science was plunged into the abyss of uncontrollable chaos.

Pic. 12. Andrew Wiles

Already three and a half centuries have passed since the first publication of the FLT and twenty-five years after it was announced that in 1995 this problem was allegedly solved by Professor Princeton University USA Andrew Wiles.⁶ However, once again it turned out this “epochal” event has nothing to do with the FLT!⁷ “The proof” of Wiles rests solely on the idea proposed by the German mathematician Gerhard Frey. This idea was rated as brilliant, but apparently only because that it was an elementary and even very common error!!!

Pic. 13. Gerhard Frey

Instead of proving the impossibility of the Fermat equation aⁿ+bⁿ=cⁿ in integers for n>2 here is proven only its incompatibility in the system with the equation y²=x(x−aⁿ)(x+bⁿ). In a similar way any nonsense can be proven. If the same work would be presented by one of the students, any of the professors would quickly bring him to clean water pointing to the obvious substitution of the subject of proof. Nevertheless, this super sensational news with great fanfare was noted in the world's leading media. The most influential newspaper of the USA “The New York Times” has been reported this right on the front page… in whole 2 years before the appearance of the “proof” itself!!! Andrew Wiles as the author of the"proof"became a member of the French Academy of Sciences and the laureate of as many as 18 of the most prestigious awards!!! To cover this momentous event, the British broadcaster BBC released an enthusiastic film and also it was invited the writer Simon Singh who published a book in 1997 titled “The Fermat's Last Theorem. The story of a riddle that confounded the world's greatest minds for 358 years”.

Pic. 15. Simon Singh

Pic. 14. “The New York Times” of 06/24/1993 with an Article About Solving the FLT Problem

If Singh independently was preparing this book, then he would have so many questions that he would not have them managed for 20 years. Of course, he was helped in every way by the very heroes-professors having glorified in the BBC film, therefore the book became a success and it is really interesting to read it even to those who know about mathematics only by hearsay. The first thing that immediately catches your eye, is the fact that in the book it was made an arithmetic error (!) and not somewhere, but in its very name! Indeed, it is well known that “the greatest minds” could not know anything about the FLT before 1670 when its wording first appeared in a book published by Fermat’s son Clément Samuel “Arithmetic” by Diophantus with comments by K. Bachet and P. Fermat (see Appendix VI Pic. 96).⁸ But then it should be not 358 but 325 years and it turns out that Singh simply did not notice the error?

However, don't rush to conclusions! This is not the book's author error and not at all accidental. These same professors vividly told Singh that supposedly back in 1637 ⁹ Fermat himself had noticed an error in his proof, but simply forgot to strike out recording of this theorem in the margins of the book. Who had invented this tale is unknown, but many scientists perceived it as a known fact and repeated time after time in their works. One can understand them because otherwise we could believe that Fermat turned out to be smarter than all of them! When Andrew Wiles said (https://www.pbs.org/wgbh/nova/article/andrew-wiles-fermat/):

“I don't believe Fermat had a proof” — this opinion was not new at all because many reputable scientists have repeated this many time. However, this is clearly against logic. It turns out that Fermat somehow managed to formulate an absolutely not obvious theorem without any reason whatsoever.¹⁰

Another contradiction in Singh’s book is a clear discrepancy between the documentary facts and the assessments of Fermat as a scientist by consultants. It is necessary to pay tribute to Singh in that he is in good faith (although not fully) outlined that part of the Fermat's works, which relates to his contribution to science and is confirmed documental. Especially it should be noted that arithmetic is called in his book"the most fundamental of all mathematical disciplines". Only one listing of Fermat's achievements in science is enough to be sure that there were only a few scientists of such a level in the entire history of science.

But if this is so, then why was it necessary to think out something that is not confirmed by any facts and only distorts the real picture? This is very similar to the desire to convince everyone that Fermat could not prove the FLT since this is allegedly confirmed by historians. But historians received information from those mathematicians who did not cope with the Fermat’s tasks and could in this way express their discontent. Hence, it's clear how appear all the arguments taken from nowhere that Fermat was an amateur scientist, arithmetic attracted him only with puzzles, which he “invented”, FLT also was by him “invented” looking at the Pythagorean equation, and his proofs he did not want to publish because fear of criticism of colleagues.

That's what they really meant! Instead of the greatest scientist and founder of number theory as well as combinatorics (along with Leibniz), analytical geometry (along with Descartes), probability theory (along with B. Pascal), wave optics theory (along with Huygens), differential calculus (along with Leibniz and Newton), whose heritage was used by the greatest scientists in the course of centuries, suddenly a “lover” of puzzles appear, who only enjoyed the fact that no one could solve them. And since arithmetic is puzzles then this most fundamental of all sciences is relegated to the level of crosswords. Such a “logic” is clearly sewn with white threads and to be convinced of this, it is enough just to point out some well-known facts.

History has not retained any evidence that during the period life and activity of P. Fermat, someone has solved at least one of his tasks.¹¹ This fact became the basis for opponents else in those times to compose all kinds of tales about him. In the surviving letters, he reported that he had already sent proofs to his respondents three times. But none of these proofs reached us because Fermat's letters recipients in eyes of posterity of course, did not want to look like they could not cope with simple tasks. Another indisputable fact is that the Fermat's personal copy of the book “Arithmetic” by Diophantus edited in 1621 with his handwritten comments in the margins, none of the eyewitnesses have ever seen!!! Well, now just a most curious picture turns out. Fermat’s critics seriously believe a witty Gascon joke that the Honorable Senator (apparently because of his lack of paper!) writes accurate and verified text of thirty-six Latin words in the book's margins, but are absolutely don't believe that he (the greatest scientist!) indeed had “truly amazing proof” of his own theorem.¹²

It is even difficult to imagine how these critics would have been amazed to find out that in fact Fermat had never dealt with the search for this proof since at that time he could not know what exactly is to be proven. But namely in the last sentence of the FLT wording, which had so much outraged them, there is a keyword directly indicated how he have solved this problem. It so happened that for centuries the science world vainly tormented itself in search of the FLT proof, but Fermat himself was never looked for it and simply had declared that he had it discovered!¹³

It is possible also to remind to opponents ingeminating about Fermat’s deliberate refusal to publish his works that for example, Descartes had received permission to publishing from Most Reverend cardinal Richelieu himself. It was impossible for Fermat and there is even a written (!!!) testimony about it (see text on P. Fermat’s tombstone: “Vir ostentationis expers… — He was deprived the possibility of publication…”. See Appendix VI Pic. 93 — 94). Nevertheless, even being in such conditions, he had prepared the publication of Diophantus’ “Arithmetic” with the addition of his 48 comments, one of which got a name the “Fermat’s Last Theorem”.

The publication was supposed to appear in honor of the historically significant event — the foundation of the French Academy of Sciences, in which preparation Fermat himself participated through the correspondence with his long-time colleague from the Toulouse parliament Pierre de Carcavy who became the royal librarian. The royal decree of the creation of the French Academy of Sciences was prepared by Carcavy and the all-powerful Finance Minister Jean-Baptiste Colbert submitting it to the signing by Louis XIV. However, the Academy of Sciences was established only in 1666 i.e. a year after the Fermat's death.

Mathematicians are very famous for how they are strict pedants, formalists and quibblers, but as soon as it comes to the FLT, all these qualities immediately disappear somewhere. Fermat's opponents ignoring well-known facts, called him either a hermit (this is a senator from Toulouse!) or a prince of amateurs (this is one of the founders of the French Academy of Sciences!), and this despite his contribution to science comparable to its importance only with a couple or triple of the most prominent scientists in the history of science!

They also did not fail sarcastically to point out that no one would have known about Fermat if the greatest mathematician of all times and peoples Leonhard Euler had not become interested in his tasks. But just this magic name has played a cruel joke with them. Their boundless belief in Euler's innovatory researches was too blind to notice that it was namely thanks to him science received such a powerful blow, from which it cannot recover up to now!

Mathematicians not only have believed Euler, but also warmly supported him that algebra is the main mathematical science, while arithmetic is only one of its elementary sections.¹⁴ Euler's idea was really excellent because his algebra, which gained new possibilities through the use of"complex numbers"was to be a most powerful scientific breakthrough that would allow not only to expand the range of numbers from the number axis to the number plane, but also to reduce the most of all calculations to solving algebraic equations. ¹⁵

The need for"complex numbers"mathematicians explained very simply. To solve absolutely any algebraic equations, you need (not so much!) to make the equation x² + 1 = 0 become solvable. ¹⁶ In Russian this is called:"Don’t sew the tail to a mare"! This equation is not at all harmless since it has nothing to do with practical tasks, but undermines the fundamentals of science very substantially. Nevertheless, the devilish temptation to create something very spectacular on empty place turned out to be stronger than common sense and Euler decided to demonstrate the new mathematical possibilities in practice.

Pic. 16. Leonhard Euler

The FLT, which Euler could not to prove, would be perfect for demonstrating the possibilities of a new wonder-algebra. However, the result turned out to be more than modest. Instead of a general proof of FLT, only one particular case for the 3rd power was proven [8, 30]. More ambitious was seemed the proof of other Fermat’s theorem about the only solution in integers of the equation y³ = x² + 2 [36] because it was a very difficult task and like FLT, none of the mathematicians could solve it. Despite the fact that the very possibility of solvability of any algebraic equation has not yet been proven, these Euler's demonstrations were perceived by hurrah. It only remained to find a solution to the problem called the “Basic Theorem of Algebra”. In 1799 the real titan of science Carl Gauss coped brilliantly with this task presenting proof even in 4 different ways!

The scientific community greeted all these"achievements"with a storm of applause while the unholy was also so glad that it is impossible to imagine.

Pic. 17. Karl Friedrich Gauss

Yeah, this was need to be seen how the whole civilized scientific world has driven itself into a dead end! It is obvious that for science, which does not rely on arithmetic, there are no reasonable restrictions and the consequences will be sad — from the dominance of algebra, arithmetic will become so difficult that witlings will call it a science for the elitist mathematicians where they can demonstrate the sharpness of their mind! But the scientists themselves unsuspecting and full of the best intentions, continued to advance science forward to new heights, but so diligently that they either inadvertently or due to a misunderstanding… simply have lost the Fermat's Golden Theorem (FGT)! But this was one of the most impressive discoveries of Pierre Fermat in arithmetic, of which he was very proud.

Pic. 18. Joseph Lagrange

It was so happened that the third in the history royal mathematician Joseph Lagrange together with his predecessor the second royal (and the first imperial!) mathematician Leonard Euler, have proven in 1772 only one special case of FGT for squares and became famous for all the world. This remarkable achievement of science was called the “Lagrange's Theorem about Four Squares”. Probably it is good that Lagrange didn’t live after two years until the moment when in 1815 still very young Augustin Cauchy presented his general proof of the FGT for all polygonal numbers. But then suddenly something terrible happened, the unholy appeared from nowhere and put his"fe"in. And here isn't to you any world fame and besides, you get complete obstruction from colleagues.

Pic. 19. Augustin Cauchy

Well, nothing can be done here, academicians did not like Cauchy and they achieved that this general proof of the FGT was ignored and did not fall into the textbooks as well as no one remembers the Gauss' proofs of 1801 for triangles and for the same squares, nevertheless in the all textbooks as before and very detailed the famous Lagrange's theorem is given. However, after Google published a facsimile of the Cauchy proof of FGT [3], it became clear to everyone why it was not supported by academics (see pt. 3.4.2).

Pic. 20. Marie-Sophie Germain

In the meantime, scientists from around the world inspired by these grand shifts, have so perked up that they wanted overcome the very FLT! They were joined by another famous woman very well known among scientists and mathematicians Marie-Sophie Germain. This talented and ambitious Mademoiselle proposed an elegant way, which was used by at once two giants of mathematical thought Lejeune Dirichlet and Adrien Legendre to prove… only one special case of FLT for the fifth power.

Another such giant Gabriel Lame managed to do the almost impossible and get proof of the highest difficulty… of another particular case of FLT for the seventh power. Thus, the whole elitist quad of the representatives from the high society of scientists was able to prove whole two (!) particular cases of FLT [6, 38].

Pic. 21. Lejeune Dirichlet

Pic. 22. Adrien Legendre

Pic. 23. Gabriel Lame

This result could have been proud since even Euler was also able to prove only two particular cases of FLT for 3rd and 4th powers. In the proof for the 4th power he has applied the descent method following exactly the recommendations of Fermat (see Appendix II). This case is especially important because its proof is valid for all even powers i.e. to obtain a general proof of FLT only odd powers can be considered.

It should be noted that namely Euler has solved (and even significantly expanded!) almost all the most difficult Fermat's tasks and if not for him, then the name Fermat alone could cause real chills to mathematicians. But just not to Sophie Germain who was not at all satisfied the situation with the unproven FLT and she even ventured to suggest that Gauss himself should take up this task! But he simply waved away her replying that the FLT is of little interest to him and such statements, which can neither be proven nor refuted can be found as many as you like.

Of course, Gauss himself would be happy to serve this lady, but if he could do this then it would not need to persuade him. For example, with the help of the “Deductions' arithmetic” developed by Gauss, the prototype of which was the “The Fermat's little theorem”, it was clearly shown how may be to solve the most difficult problems of arithmetic effectively. In particular, only Gauss managed to find a solution to the Fermat's task of calculating two the only squares, the sum of which is a given prime number of types 4n+1 [11, 25].

A characteristic feature of Gauss is his dislike for dubious innovations. For example, he could hardly imagine himself the creator of the geometry of curved spaces. But when he established that such geometry could take place and not contain contradictions, he was very puzzled by this. He was sure that his find could not be of practical use due to the absence of any real facts confirming something like that. However, he quickly found a good way out — he just helped to publish this discovery to his Russian colleague Nikolai Lobachevsky and have done it so skillfully that no one was even surprised when a Russian professor and rector of Kazan University have published a work on non-Euclidean geometry… in Berlin and in German! In the future, Gauss' doubts were confirmed. Followers appeared and flooded science with a whole bunch of similar"discoveries".

Despite the fact that with his proof of the “Basic Theorem of Algebra” Gauss supported Euler in promoting his idea of using “complex numbers”, he did not find any other opportunities for progress in this direction. And what Euler showed, he was also not impressed. Moreover, even modern science at all can nothing offer anything on the use of “complex numbers”. But the sea of all kinds of “scientific” works, studies and textbooks on this theme is clearly inadequate with its true value. Gauss felt that something was amiss with these “numbers” and that it would not end well, therefore in that direction he did not work.

Pic. 24. Ernst Kummer

Thunder struck in 1847 when at a meeting of members of the French Academy of Sciences Gabriel Lame and Augustin Cauchy reported that their FLT proofs was ready for consideration at the competition. However, when in order to identify the winner, it was already possible to open received from them the sealed envelopes, the German mathematician Ernst Kummer having put all scientists on the sinful earth. In his letter it was reported that the FLT proof on the basis of “complex numbers” is impossible due to the ambiguity of their decomposition into prime factors.¹⁷

Here you have got what you want! These very “complex numbers” are not any numbers!!! And one could notice finally, after arithmetic was knocked from under science, it hangs in the air having no solid foundation. And the mistakes of the greats in their consequences are also extreme and they begin to break down a science so much that, instead of a holistic system of knowledge, it creates a bunch of unrelated fragments.

If it so happened, then else in 1847, these very “Complex numbers” had to be solemnly buried with all the honors. But with this matter somehow did not work out at all and the restless souls of the long-dead theories turn out to be so tenacious that they cannot be expelled from textbooks and professorial lectures by any means. They will wander through different books and reference books whose authors will be completely unaware of how much their works depreciates from this useless ballast.

In the mentioned book of Singh is well shown as the ambiguity of decomposing compound integers into prime factors makes it impossible to construct logical conclusions in proofs and it also was said that the unambiguity theorem for such a decomposing for natural numbers was given in “Euclidean Elements”. The specific book and location of the theorem is not specified; therefore, it is rather difficult to find the necessary text, but it really turned out to be so.¹⁸

“Euclidean Elements"is a very old book with archaic terminology, in which this extremely important for science theorem was somehow lost and it was simply forgotten about it. The first to discover the loss was Gauss. He formulated it again and gave proof, which contained a surprisingly simple and even childish error, where as an argument used exactly what needs to be proven (see pt. 3.3.1).

But this is not an ordinary theorem, all science holds on it! And what about Euclid? Oh my God! In fact, his proof is the same as that of Gauss i.e. wrong!!! Tell it to someone, so they will not believe! Three giants of science are stumbled on the same place!

Pic. 25. Euclid

Then it turns out that this whole science is fake and now, thanks to Singh’s book and despite all the good intentions of its author, this terrifying FLT, which now even in theory has become completely unprovable, was so furious that like a true monster, in one fell swoop have devalued all the age-old works of scientists! And yet they live in not fabulous, but in the real kingdom of crooked mirrors, what about they themselves don’t know anything.

The fiasco being by academicians Cauchy and Lame did not result in the rejection use of the surrogates of numbers in science especially after Kummer who had crushed their works, found a way to prove FLT (with a little modernization) for any particular case. Before the final victory over the FLT only a last step remained — to obtain a single common proof. Since then 170 years have passed, but nothing was changed. Supported in due time by the Euler's genius"complex numbers"are still presented today as a kind of extension the notion of number. This looks very impressive and solid, but still requires a clear definition of the very notion of number, however just with this deal are very bad.

Students intuitively feeling that they are being tortured in vain by nonsenses about some non-existent numbers, suddenly have a question: “What is a number?” They never come to mind that not a single professor could not answer this question even if he has reread everything that is in mathematics. One of them even could not bear the mocking hints and had published a whole book called “What is a number?” [13, 29]. In it, he has written so many whatnots that students have very well understood — such a question it’s better not to ask.

Pic. 26. Francis Viète

Meanwhile, scientists continued to move science forward, not bothering with such trifles as the essence of the notion of number. So, they created a whole bunch of new algebras taking advantage of the fact that there were no obstacles along the way. But they were not a continuation of what was a real one, the founder of which was the first royal mathematician François Viète served as an advisor to the court of the French king Henry III. But if these new algebras are special, then their terminology and bases are also special.

So, little by little in the science began to form a particular bird language understandable only to the authors of these most innovative developments. It even reached the point where mathematical societies creating a science only for themselves began to appear and in addition to this, the newest numbers appeared out of nothing: “hypercomplex”, “quaternions”, “octonions” etc. But the impression from the new achievements sometimes was spoiled from the same mare tail,¹⁹ which from somewhere appeared again. Getting this tail in the face is not very pleasant, but this is already the costs of a profession. In an effort to get away from such costs, a brilliant way out of the difficulties with the definition the essence the notion of a number was found. Scientists have finally grasped that it needs to be derived from other simpler notions, for example, such as the notion “set”. Everything turned out so simple, a set is that what is a lot. Well, is it not clear? However, it was found out again that one cannot do without empty set and in this case, it would see like nothing, and the question again arises, so what is a set number or not?

Georg Cantor has developed his theory of sets, which other mathematicians such as, for example, Henri Poincaré, called all sorts of bad words and did not want to admit at all. But suddenly unexpected for everyone the respectable"Royal Society of London"(the English Academy of Sciences) in 1904 decided to award Cantor with its medal. So, it turns out that here is the point, where the fates of science are decided!²⁰

Pic. 27. Georg Cantor

And everything would be fine, but suddenly another trouble struck again. Out of nowhere in this very theory of sets insurmountable contradictions began to appear, which are also described in great detail in Singh’s book. In the scientific community everyone immediately was alarmed and began to think about how to solve this problem. But it has rested as on the wall and in no way did not want to be solved. Everyone was somehow depressed, but then they yet cheered up again.

It was so happened because now David Hilbert himself got down to it, the great mathematician that first solved the very difficult Waring problem, which has a direct relationship to the FLT. ²¹ It is also curious that Hilbert repeated Euler's experiment apparently inspired by the FLT problem. It seems that at some point Euler began to have doubts that the FLT is generally provable and he assumed the equation a⁴+b⁴+c⁴=d⁴ also like Fermat’s equation aⁿ+bⁿ=cⁿ for n>2 in integers is unsolvable, but in the end it turned out that he was wrong.²²

Pic. 28. David Hilbert

Following the example of Euler on the eve of the 20th century, Hilbert offered to the scientific community 23 problems, which according to his assumption, are unlikely to be solved in the foreseeable future. Nevertheless, Hilbert's colleagues coped with them rather quickly, while Euler’s hypothesis has held almost until the 21st century and was only refuted with the help of computers, what is also described in Singh’s book. So, the suspicion that the FLT was merely an assumption of its author has lost any reason.

Hilbert had not cope with overcoming contradictions in set theory and could not do it because this problem is not at all mathematical, but informational one, so computer scientists should solve it sooner or later and when this happened, they are surprisingly very easily (and absolutely true) found a solution just forbidding closed chains of links.²³ It is clear Hilbert could not know about it then and decided that the most reliable barrier to contradictions can be provided with the help of axioms. But axioms cannot be created on empty place and must come out of something and this something is a number, but what it is, no one can explain this not then nor now.

A brilliant example of what can be created with axioms is given in the same book of Singh. The obvious incident with the lack of a clear formulation to the notion of a number can accidentally spoil any rainbow picture and something needs to be done with it. It gets especially unpleasant with the justification of the “complex numbers”. Perhaps this caused the appearance in the Singh’s book of Appendix 8 called “Axioms of arithmetic”, in which 5 previously known axioms relating to a count are not mentioned at all (otherwise the idea will not past), while those that define the basic properties of numbers are complemented and a new axiom appears so that it must exist the numbers n and k, such that n+k=0 and then everything will be in the openwork!

Of course, Singh himself would never have guessed this. It is clearly visible here the help of consultants who for some reason forgot to change the name of the application since these are no longer axioms of arithmetic because already nothing is left of it.²⁴ The school arithmetic, which for a long time barely kept on the multiplication table and the proportions, is now completely drained. Instead it, now there is full swing mastering of the calculator and computer. If such “progress” continues further, then the transition to life on trees for our civilization will occur very quickly and naturally.

Against this background a truly outstanding scientific discovery was made in Wikipedia, which simply has no equal in terms of art and the scale of misinformation. For a long time, many people thought that there are only four actions of arithmetic, these are addition and subtraction, multiplication and division. But no! There are also exponentiation and… root extraction (???). The authors of the articles given us this"knowledge"through Wikipedia clearly blundered because extracting the root is the same exponentiation only not with the integer power, but with fractional one. No of course, they knew about it, but what they didn’t guess was that it was they who copied this arithmetic action at Euler himself from that very book about the wonder-algebra²⁵.

The correct name of the sixth action of arithmetic is logarithm i.e. calculating the power index (x) for a given power number (y) and basis of a power (z) i.e. from y=z^x follows x=log_zy. As in the case with the name of the Singh’s book, this error is not at all accidental since no one really worked on logarithms as part of the arithmetic of integers. If this happens someday, then not earlier than in some five hundred years! But as for the action with power numbers, the situation here is not much better than with logarithms. If multiplication and dividing of power numbers as well as exponentiation a power number to a power, do not present any difficulties, but the addition of power numbers is still a dark forest even for professors.

The clarification in this matter begins with the FLT, which states that the sum of two power integer with the same power index greater than the second, cannot be an integer with the same power index. In this sense, this theorem is not at all any puzzle, but one of the basic propositions that unequivocally (!) regulates the addition of integer powers, therefore, it is of fundamental importance for science.²⁶ The fact that the FLT has not yet been proven, indicates only the state of current science, which is falling apart right before our eyes. Science cannot even imagine that if the proof from Fermat himself came to us, it would have been long ago taught in secondary school.

Many people of course, will perceive it as a fairy tale, but only the completely blind ones may not notice that behind all this absurd and awkward history with the FLT, clearly and openly ears of the unholy stick so out, that he was enough to deprive human civilization of access to Fermat's works on arithmetic, so it immediately turned out to be completely disoriented. Instead of developing science they began being vigorously to destroy it and even with very good intentions. But a special zeal in people appears when they have the material stimulus.

Pic. 29. Andrew Beal

Texas entrepreneur Andrew Beal²⁷ had proposed his conjecture, the proof of which allegedly could lead to a very simple proof of the FLT. Since for the solution of this problem it was proposed first $ 5 thousand, then $ 100 thousand, and from 2013 — a whole million, then naturally it appeared many willing people who began diligently this task to solve. However, in the conditions when arithmetic has long ceased to be the primary basic of all knowledge and still does not know, what is a number, everything turned upside down i.e. one amateur enthusiast was able to set on the ears the whole official science and so, that it had in fact already acknowledged the experience of Baron Munchhausen lifting himself up, taking himself by his collar, wherewith science did not even try at least to conceal its own insolvency (see pt. 4.5).

By working in the intense and tireless search for the FLT proof, it has never even occurred to anyone to search for Fermat’s manuscripts with layouts and calculations, without which he could not do²⁸. However, again from Singh’s book we learn that such an idea came to Euler who asked his friend living in Lausanne (a city not far from Toulouse) to look for at least a little piece of paper with Fermat’s instructions to the FLT proof. But nothing was found, however, they were looking for what we do not really need! It was necessary to look for a cache!!!

Here is the new puzzle, which is not easier! What else kind of cache?… Oh yes! The fact is that only those Fermat's works remained, which he itself had already prepared for publication since otherwise they would hardly have been published. But all the working manuscripts for some reason has disappeared. It looks very strange and it is possible that they can still be kept in the cache, which Fermat has equipped to store the material evidence necessary for him to work as a senator and high-ranking judge. It was quite reasonable to keep calculations and proofs there, since Fermat’s scientific achievements could significantly damage his main work if they were made public before the establishment of the French Academy of Sciences.²⁹

If we could somehow look into this cache, what will we see there? To begin with, let's try to find some simple tasks there. For example, the one that Fermat could offer today for secondary school students:

Divide the number xⁿ−1 by the number x−1, or the number x²ⁿ−1 by

the number x±1, or the number x²ⁿ⁺¹+1 by the number x+1.

It is obvious that students with the knowledge of solving such a task will be simply a head over the current students who are trained in the methods of determining the divisibility by only some small numbers. But if they else know a couple of the Fermat's theorems, they can easily solve also the more difficult problem:

Find two pairs of squares, each of which adds up to the same number

in the seventh power, for example,

221⁷=151114054²[phone]²=82736654 ²[phone]²

Compared to the previous task where calculations are not needed at all, in solving this task, even with a computer calculator you have to tinker with half an hour to achieve a result, while apart from understanding the essence of the problem solution, you need to show a fair amount of patience, perseverance and attention. And who understands the essence of the solution, will be able to find other solutions to this problem.³⁰

Of course, such tasks can cause a real shock to today's students and especially to their parents who will even demand not to “dry the brains” of children. But if children's brains are not filled with elementary knowledge and not trained by solving the corresponding tasks, they will wither by themselves. This is proven by the statistics of the steady decline in today's students IQ compared with their predecessors. Really in fact, the above tasks are only a warm-up for the young generation, but children could produce a real furor for mathematicians offering them some simple Fermat's theorems about magic numbers (see Pt. 4.4.). And this is else a big question, could these theorems being solved by today's professors or will they again need some three hundred years and the story with the FLT will repeat? However, the chances of them in contrast to previous times, are very high because magic numbers are a direct consequence of the same “truly amazing” proof of the FLT, about the existence of which we have direct written evidence from Fermat himself.

Reconstruction of this proof was briefly published as early as 2008 [30], but the unholy was on the alert and presented this event so, that modern science disoriented by the false notion that the problem was solved long ago, has not paid on this any attention. However, all secret sooner or later becomes clear and the decisive word in spite of everything, still remains for science. The question now is only when this science will finally awaken and comes to his senses. The longer it will be in a blissful state of oblivion, the sooner the terrible events will come that already now beginning to shake our world like never before.

In order for science to win a well-deserved victory over the gloom of ignorance and mass disinformation, which are triumphant today, it needs very little. For the beginning it is necessary simply to search for the very cache, in which such secrets of science are hidden, that have not lost their relevance for three and a half centuries.³¹ Even if the papers found in the cache will be unreadable, the very fact of the existence of the cache will be evidence that science is moving in the right direction and the results will not be long in coming.

We already did something in this direction when we restored the FLT recording in the margins of Diophantus 'Arithmetic' (see pic. 5 and the translation in the end of Pt. 1). Now, by all means, we need to get a complete picture of the whole sequence of events that led to the discovery of the FLT in its final wording published in 1670. It will not be easily at all, but since we got involved in this story, now we have nowhere to retreat and we will strain all our forces to achieve the aim. Fortunately, for this we have all the opportunities granted to us from above to get the coveted access to the cache of the Toulousean senator.

Примечания

6

It was a truly grandiose mystification, organized by Princeton University in 1995 after publishing in its own commercial edition"Annals of Mathematics"the “proof” of FLT by A. Wiles and the most powerful campaign in the media. It would seem that such a sensational scientific achievement should have been released in large numbers all over the world. But no! Understanding of this text is available only to specialists with appropriate training. Wow, now even that, which cannot be understood, may be considered as proof! However, for fairness it should be recognized that even such an overtly cynical mockery of science, presented as the greatest"scientific achievement"of the luminaries of Princeton University, cannot be even near to the brilliant swindle of their countrymen from the National Space Administration NASA, which resulted that the entire civilized world for half a century haven’t any doubt that the American astronauts actually traveled to the moon!

7

The “proof”, which A. Wiles prepared for seven years of hard work and published on whole 130 (!!!) journal pages, exceeded all reasonable limits of scientific creativity and of course, him was awaiting inevitable bitter disappointment because such an impressive amount of casuistry understandable only to its author, neither in form nor in content is in any way suitable to present this as proof. But here the real wonder happened. Suddenly, the almighty unholy himself was appeared! Immediately there were influential people who picked up the"brilliant ideas"and launched a stormy PR campaign. And here is your world fame, please, many titles and awards! The doors to the most prestigious institutions are open! But such a wonder even for the enemy not to be wish because sooner or later the swindle will open anyway.

8

If this book was published during the life of Fermat, then he would simply be torn to pieces because in his 48 remarks he did not give a proof of any one of his theorems. But in 1670 i.e. 5 years after his death, there was no one to punish with and venerable mathematicians themselves had to look for solutions to the problems proposed by him. But with this they obviously had not managed and of course, many of them could not forgive Fermat of such insolence. They were also not forgotten that during his lifetime he twice arranged the challenges to English mathematicians, which they evidently could not cope with, despite his generous recognition of them as worthy rivals in the letters they received from Fermat. Only 68 years after the first publication of Diophantus'"Arithmetic"with Fermat's remarks, did the situation at last get off the ground when the greatest science genius Leonard Euler had proven a special case of FLT for n=4, using the descent method in exact accordance with Fermat's recommendations (see Appendix II). Later thanks to Euler, there was received solutions also of the other tasks, but the FLT had so not obeyed to anyone.

9

In pt. 2-30 of the letter Fermat to Mersenne, the task is set:

“Find two quadrate-quadrate, the sum of which is equal to a quadrate-quadrate or two cubes, the sum of which is a cube” [9, 36]. The dating of this letter in the edition by Tannery is doubtful since it was written after the letters with a later dating. Therefore, it was most likely written in 1638. From this it is concluded that the FLT is appeared in 1637??? But have the FLT really such a wording? Even if these two tasks are special cases of the FLT, how it can be attributed to Fermat what about he could hardly even have guessed at that time? In addition, the Arabic mathematician Abu Mohammed al Khujandi first pointed to the insolubility of the problem of decomposing a cube into a sum of two cubes as early else the 10th century [36]. But the insolvability of the same problem with biquadrates is a consequence of the solution of the problem from pt. 2-10 of the same letter:"Find a right triangle in numbers whose area would be equal to a square."The way of proving Fermat gives in his 45th remark to Diophantus' “Arithmetic”, which begins like this: “If the area of the triangle were a square then two quadrate-quadrates would be given, the difference of which would be a square.” Thus, at that time, the wording of this problem and the approach to its solution were very different even from the particular case of FLT.

10

In order no doubts to appear, attempts were made to somehow “substantiate” the fact that Fermat could not have the proof mentioned in the original of FLT text. See for example, https://cs.uwaterloo.ca/~alopez-o/math-faq/node26.html (Did Fermat prove this theorem?). Such an"argument"to any of the sensible people related to science, it would never come to mind because it cannot be convincing even in principle since in this way any drivel can be attributed to Fermat. But the initiators of such stuffing clearly did not take into account that this is exactly evidence of an organized and directed information campaign on the part of those who were interested in promoting Wiles’ “proof”.

11

An exception is one of the greatest English mathematicians John Wallis (see pt. 3.4.3).

12

Obviously, if it come only about the wording of the FLT, it would be very unwise to write it in the margins of the book. But Fermat’s excuses about narrow fields are repeated in other remarks for example, in the 45^th, at the end of which he adds: “Full proof and extensive explanations cannot fit in the margins because of their narrowness” [36]. But only one this remark takes the whole printed page! Of course, he had no doubt that his Gascon humor would be appreciated. When his son, Clement Samuel who naturally found a discrepancy in the notes prepared for publication, was not at all surprised by this since it was obvious to him that right after reading the book it was absolutely impossible to give exact wording of tasks and theorems. The fact that this copy of Diophantus’ “Arithmetic” with Fermat's handwritten notes didn’t come to us suggests that even then this book was an extremely valuable rarity, so it could have been bought by another owner for a very high price. And he was of course not so stupid to trumpet about it to the whole world at least for his own safety.

13

The text of the last FLT phrase: “I have discovered a truly amazing proof to this, but these margins are too narrow to put it here”, obviously does not belong to the essence of the theorem, but for many mathematicians it looks so defiant that they tried in every way to show that it's just empty a Gascon boasting. At the same time, they did not notice neither humor about the margins nor the keyword “discovered”, which is clearly not appropriate here. More appropriated words here could be, say, “obtained” or “founded”. If Fermat’s opponents paid attention to this, it would become clear to them that the word “discovered” indicates that he received the proof unexpectedly by solving the Diophantus' task, to which a remark was written called the FLT. Thus, mathematicians have unsuccessfully searched during the centuries for FLT proof instead of looking for a solution to the Diophantus' task of decomposing a square into the sum of two square. It seemed to them that the of Diophantus' task was clearly not worth their attention. But for Fermat it became perhaps the most difficult of all with it he has worked on, and when he did cope with it, then received the discovery of the FLT proof as a reward.

14

It is curious that the Russian-language edition this fundamental work of Euler was published in 1768 under the title"Universal Arithmetic"although the original name"Vollständige Anleitung zur Algebra"should be translated as the Complete Introduction to Algebra. Apparently, translators (students Peter Inokhodtsev and Ivan Yudin) reasonably believed that the equations are studied here mainly from the point of view of their solutions in integers or rational numbers i.e. by arithmetic methods. For today's reader this 2-volume edition is presented as a Chinese literacy because along with the highly outdated Russian language and spelling, there is simply an incredible number of typos. It is unlikely that today's RAS as the heiress of the Imperial Academy of Sciences, which published this work, understands its true value, otherwise it would have been reprinted a long time ago in a modern and accessible form.

15

Here there is an analogy between algebra and the analytic geometry of Descartes and Fermat, which looks more universal than the Euclidean geometry. Nevertheless, Euclidean arithmetic and geometry are the only the foundations, on which algebra and analytical geometry can appear. In this sense, the idea of Euler to consider all calculations through the prism of algebra is knowingly flawed. But his logic was completely different. He understood that if science develops only by increasing the variety of equations, which it is capable to solve, then sooner or later it will reach a dead end. And in this sense, his research was of great value for science. Another thing is that their algebraic form was perceived as the main way of development, and this later led to devastating consequences.

16

Just here is the concept of a “number plane” appears, where real numbers are located along the x axis, and imaginary numbers along the y axis i.e. the same real, only multiplied by the “number” i = √-1. But along that come a contradiction between these axes — on the real axis, the factor 1ⁿ is neutral, but on the imaginary axis no, however this does not agree with the basic properties of numbers. If the “number” i is already entered, then it must be present on both axes, but then there is no sense in introducing the second axis. So, it turns out that from the point of view of the basic properties of numbers, the ephemeral creation in the form of a number plane is a complete nonsense.

17

According to the Basic theorem of arithmetic the decomposition of any natural number into prime factors is always unambiguous, for example, 12=2×2×3 i.e. with other prime factors this number like any other, is impossible to imagine. But for “complex numbers” in the general case this unambiguity is lost for example, 12=(1+√–11)×(1+√–11)=(2+√–8)×(2+√–8) In fact, this means the collapse of science in its very foundations. However, the generally accepted criteria (in the form of axioms) what can be attributed to numbers and what is not, as there was not so still is not.

18

The theorem and its proof are given in “The Euclid's Elements” Book IX, Proposition 14. Without this theorem, the solution of the prevailing set of arithmetic problems becomes either incomplete or impossible at all.

19

Soviet mathematician Lev Pontryagin showed these “numbers” do not have the basic property of commutativity i.e. for them ab ≠ ba [34]. Therefore, one and the same such “number” should be represented only in the factorized form, otherwise it will have different value at the same time. When in justification of such creations scientists say that mathematicians have lack some numbers, in reality this may mean they obviously have lacked a mind.

20

If some very respected public institution thus encourages the development of science then what one can object? However, such an emerging unknown from where the generosity and disinterestedness from the side of the benefactors who didn’t clear come from, looks somehow strange if not to say knowingly biased. Indeed, it has long been well known where these “good intentions” come from and whither they lead and the result of these acts is also obvious. The more institutions there are for encouraging scientists, the more real science is in ruins. What is costed only one Nobel Prize for"discovery"of, you just think… accelerated scattering of galaxies!!!

21

Waring's problem is the statement that any positive integer N can be represented as a sum of the same powers x_iⁿ, i.e. in the form N = x₁ⁿ + x₂ⁿ +… + x_kⁿ. It was in very complex way first proven by Hilbert in 1909, and in 1920 the mathematicians Hardy and Littlewood simplified the proof, but their methods were not yet elementary. And only in 1942 the Soviet mathematician Yu. V. Linnik has published arithmetic proof using the Shnielerman method. The Waring-Hilbert theorem is of fundamental importance from the point of view the addition of powers and does not contradict to FLT since there are no restrictions on the number of summands.

22

A counterexample refuting Euler’s hypothesis is 95800⁴ + 217519⁴ + 414560⁴ = 422481⁴. Another example 2682440⁴[phone]⁴[phone]⁴=20615673⁴. For the fifth power everything is much simpler. 27⁵+84⁵+110⁵+133⁵=144⁵. It is also possible that a general method of such calculations can be developed if we can obtain the corresponding constructive proof of the Waring's problem.

23

Of course, this does not mean that computer scientists understand this problem better than Hilbert. They just had no choice because closed links are looping and this will lead to the computer freezing.

24

The axiom that the sum of two positive integers can be equal to zero is clearly not related to arithmetic since with numbers that are natural or derived from them this is clearly impossible. But if there is only algebra and no arithmetic, then also not only a such things would become possible.

25

It is curious that even Euler (apparently by mistake) called root extraction the operation inverse to exponentiation [8], although he knew very well that this is not so. But this is no secret that even very talented people often get confused in very simple things. Euler obviously did not feel the craving for the formal construction of the foundations of science since he always had an abundance of all sorts of other ideas. He thought that with the formalities could also others coped, but it turned out that it was from here the biggest problem grew.

26

This is evident at least from the fact, in what a powerful impetus for the development of science were embodied countless attempts to prove the FLT. In addition, the FLT proof, obtained by Fermat, opens the way to solving the Pythagorean equation in a new way (see pt. 4.3) and magic numbers like a+b-c=a²+b²-c² (see pt. 4.4).

27

In the Russian-language section of Wikipedia, this topic is titled"Гипотеза Била". But since the author’s name is in the original Andrew Beal, we will use the name of the “Гипотеза Биэла” to avoid confusion between the names of Beal (Биэл) and Bill (Бил).

28

In a letter from Fermat to Mersenne from 06/15/1641 the following is reported: “I try to satisfy Mr. de Frenicle’s curiosity as completely as possible… However, he asked me to send a solution to one question, which I postpone until I return to Toulouse, since I am now in the village where I needed would be a lot of time to redo what I wrote on this subject and what I left in my cabinet” [9, 36]. This letter is a direct evidence that Fermat in his scientific activities could not do without his working recordings, which, judging by the documents reached us, were very voluminous and could hardly have been kept with him on various trips.

29

If Fermat would live to the time when the Academy of Sciences was established and would become an academician then in this case at first, he would publish only problem statements and only after a sufficiently long time, the main essence of their solution. Otherwise, it would seem that these tasks are too simple to study and publish in such an expensive institution.

30

To solve this problem, you need to use the formula that presented as the identity: (a²+b²)×(c²+d²)=(ac+bd)²+(ad−bc)²=(ac−bd)²+(ad+bc)². We take two numbers 4 + 9 = 13 and 1 + 16 = 17. Their product will be 13×17 = 221 = (4 + 9) × (1+16) = (2×1 + 3×4)² + (2×4 − 3×1)² = (2×1 − 3×4)² + (2×4 + 3×1)² = 14² + 5² = 10² + 11²; Now if 221⁶ = (221³)² = 10793861²; then the required result will be 221⁷ = (14² + 5²)×10793861² = (14×10793861)² + (5×10793861)² = 151114054² + 53969305² = (10² + 11²)×10793861²=(10×10793861)² + (11×10793861)²=107938610² + 118732471²; But you can go also the other way if you submit the initial numbers for example, as follows: 221² = (14² + 5²)×(10² + 11²) = (14×10 + 5×11)² + (14×11 − 5×10)² = (14×10 − 5×11)² + (14×11+5×10)² = 195² + 104² = 85² + 204²; 221³ = 221²×221 = (195² + 104²)×(10² + 11²) = (195×10 + 104×11)² + (195×11 − 104×10)² = (195×10 − 104×11)² +(195×11 + 104 × 10)² = 3 094² + 1105² = 806² + 3185²; 221⁴ = (195² + 104²)×(85² + 204²) = (195×85 + 104×204)² + (195×204 − 85×104)² = (195×85 − 104×204)² + (195×204 + 85×104)² = 37791² + 30940² = 4641² + 48620²; 221⁷ = 221³×221⁴ = (3094² + 1105²)×(37791² + 30940²) = (3094×37791 + 1105×30940)² + (3094×30940 − 1105×37791)² = (3094×37791 − 1105×30940)² + (3094×30940 + 1105×37791)²; 221⁷ = 151114054² + 53969305² = 82736654² + 137487415²

31

If Fermat's working notes were found, it would turn out that his methods for solving tasks are much simpler than those that are now known, i.e. the current science has not yet reached the level that took place in his lost works. But how could it happen that these recordings disappeared? There may be two possible versions. The first version is being Fermat’s cache, which no one knew about him. If this was so, there is almost no chance it has persisted. The house in Toulouse, where the Fermat lived with his family, was not preserved, otherwise there would have been a museum. Then there remain the places of work, this is the Toulouse Capitol (rebuilt in 1750) and the building in the city of Castres (not preserved) where Fermat led the meeting of judges. Only ghostly chances are that at least some walls have been preserved from those times. Another version is that Fermat’s papers were in his family’s possession, but for some reason were not preserved (see Appendix IV, year 1660, 1663 and 1680).