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=zx follows x=logzy. 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.