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.

1. The Greatest Phenomenon of Science

Usually, the science's image is represented as an ordered system of knowledge about everything that can be observed in the world around us. However, this image is illusory and in fact there is not any orderliness in science since it is formed not by the development of knowledge from the simple to the complex, but only by the historical process of the emergence of new theories. The classic example is the Descartes — Fermat analytic geometry, where compared with Euclidean geometry, science sees only an analytic-friendly representation of numerical functions in a coordinate system, but does not evaluate the qualitative transition from naturalized elements (point, line, surface, etc.) to numbers.¹

It would seem that this is so insignificant that it cannot have any consequences, but ironically, it was after the expansion of the numerical axis to the numerical plane, when science was hopelessly compromised, because it suddenly became clear that such a representation of numbers does not obey to the Basic theorem of arithmetic that the decomposing of an integer into prime factors is always unique. But then a corresponding conclusion should be made that no any numerical plane exists and everything connected with it should be written off to the archive of history.

But it’s really impossible! If there is no orderliness in science, then there is no reason to link new knowledge to earlier ones. Therefore, it is not at all news to the world of scientists that for the numerical plane the Basic theorem of arithmetic is not acted. This was known a century and a half ago and it never even occurred to anyone to abandon this idea. During this time, so much has been done that it’s so easily to take it all and throw away is in no way possible because many “experts” with their “scientific” research can lose their jobs and all monographs, reference books and textbooks on this theme will at once turn into tons of waste paper.²

Yes, not one of the scientists can be surprised by the fact that the Basic theorem of arithmetic is not acted, because they have already accustomed not only to such things. But they will be very surprised, when they know that nobody can prove BTA so far! All the “proofs” of this theorem in textbooks and on the Internet are either clearly erroneous or not convincing. But then it turns out that on the one hand, science deprives itself legitimacy since it does not recognize the Basic theorem, on which it itself holds, but on the other hand, it throughout all its history simply was not aware of the fact that it has no proof of this theorem.³

And what now to do? Can this blatant fact be perceived otherwise as the degradation of science in its very foundations? To some people such a conclusion may seem too categorical, but unfortunately for current science, this is even very mildly said. What a marvel, some theorem doesn’t act? And what about when the law of conservation of energy doesn’t act? Current astrophysics simply does not present itself without the “big bang theory”, according to which all the galaxies in the Universe are flowing away like fuzz. And such a crazy phantasmagoria is quite seriously presented today as one of the greatest"scientific"achievements, and fig leaves like"hidden energy"and"dark matter"easily cover the problems with the notorious conservation laws.

Against the background of the truly outstanding achievements of science there is no doubt that this virus of dark misfortune, which penetrated into its very foundations, could not have emerged from nothing and was clearly introduced from the outside. The malicious nature of the virus is disclosed by the fact that it always hides under the guise of"good intentions."And if that is so, then the task of getting rid of the misfortune is simplified because these are just the intrigues of the unholy, from which the real science always had sufficient reliable immunity.

But for this particular virus this immunity began to act in a very special way. Suddenly out of nowhere, there appeared a simple-looking task called “The Fermat’s Last Theorem” (FLT), which no one could prove despite the promised bonuses and honors. It simply scoffed at everyone who tried to find a solution regardless of whether it was an ambitious candidate for the prize or the greatest scientist. With the FLT many scientists were even afraid to deal in order not inadvertently to tarnish their reputation.

This fascinating game with a knowingly failure result dragged on centuries and in the end, everyone was so tormented that it was necessary somehow this problem try to close. Very serious people made a decision — the problem is to be solved and bonuses are to be paid. No sooner said than done. However, what happened next will be told in the next part of our work. But it will be only a preamble because in order to penetrate the essence of this amazing phenomenon we will have to come back in the past in some unusual way. And then as a result of our research, it will turns out that this task was solved long ago in the 17th century when Louis XIV the king sun began to rule in France and two Gascons faithfully served him, one of them is the well-known from novel A. Dumas is the royal musketeer Monsieur D'Artagnan and the other is his same age and countryman Senator from Toulouse Monsieur de Fermat.

The history did not preserve for us in writing everything that would be especially interesting to us, therefore, nothing remains, but to try to restore some events at that in a very unusual way what about we will also more tell. However, it is well known that this senator during his lifetime became famous for offering simple-looking arithmetic tasks to noble grandees, which for some reason no one could solve. But apparently, he didn't had time (or even perhaps he didn't want) to tell anyone about that wonderful and non-proven until now theorem therefore it is also often called the “the Fermat’s Last theorem”.

Especially curious is the fact that not a single piece of paper has been preserved from the manuscripts of his scientific works on arithmetic and even those that were published after his death. The only exceptions were letters collected from different respondents. This strange fact indicates that some amazing and even incredible course of events took place, which led to such a situation and the establishment of only this fact alone significantly changes the whole picture, which presented to researchers so far.

They even believed that Fermat could not have a proof of his Last theorem and justified it with all sorts of arguments. But then they needed to be consistent and insist that Fermat also could not solve all other his tasks since for his justification he has not left us any explanation. But if they were solved by such giants of science as, say, Euler or Gauss, well, then it is quite another thing and we could assume that Fermat also has solved them. But if even they failed, then science in no way cannot afford to trust words that look like bluster.

In our research we will go the other way and we will proceed from the fact that the proof of Fermat’s Last theorem, without any doubt, should have been written down on paper at least in a sketch version. But if this is so, then where could it have disappeared moreover along with all the other papers? The answer to this question can shed light on the healing of the above-mentioned misfortune, which led to the fact that for unknown reasons this very proof for as much as three and a half centuries has become not only an unsolvable problem, but also a real stumbling block for science.

The riddles that we now have to explore seem at first as an accidental collision of all kinds of large and small stories, but these seemingly intricate events have their own rather rigid logic. It so happened that Fermat’s life and activities coincided with a turning point in history when a slow and very painful transition to the Renaissance took place after a long period of terrifying oppression by the Inquisition, which did not tolerate advanced scientific thought and have organized in France mass destruction of Protestant-Huguenots by Catholics.

Taking into account this circumstance, it is possible to explain such facts and events that from the point of view of a later time look as very strange and not able to understand. In particular, it should be noted that in those times, especially for people of ignoble origin, it would be very dangerous to have at home even completely harmless notes with formulas and calculations that could be interpreted as a very dangerous for their owners’ recordings of heretical content.

Pierre's Father Dominique Fermat was a wealthy merchant, but did not have a noble title. In 1601 his son Pierre was born, about which there is an entry in the church book, but his mother Françoise Cazeneuve and her child died not having lived after giving birth to three years. If the child had survived, then without a noble origin, he would have no chance of becoming a senator let alone a great scholar. And when after the loss of his first wife, Dominique married Claire de Long having noble roots, then this ensured a very opportunity that the future celebrity would appear [16].

Pierre Simon de Fermat was born not in 1601 as it was believed until now, but in 1607 (or in 1608) [1] in the little town of Beaumont-de-Lomagne near Toulouse. From childhood he stood out for such talent that Dominique Fermat did not spare the funds for his education and sent him to study first in Toulouse (1620 — 1625) and then in Bordeaux and Orleans (1625–1631). Pierre did not only study well, but also showed brilliant abilities that together with his mother’s kinship and financial support from his father, gave him every opportunity to get a best education as a lawyer.

During his studies the young future Senator Pierre Fermat was very keen on reading scientific literature and was so inspired by the ideas of great thinkers that he also himself felt a desire for scientific creativity. In order to learn more about what particularly interested him, he had mastered five languages⁴ and read with enthusiasm the works from the classics of that time. As a result, he deservedly received the highest education that just was possible in those times and deep down he cherished the dream of being able to continue work in the field of science.

If the support of Pierre Fermat’s career had ended on that, then there could be no question of a future senator since in those times even simple lawyer activity demanded the highest royal deigning. From this it becomes clear why the decisive step in Pierre’s parental care was his marriage in 1631 to Louise de Long, who was a distant relative (the fourth cousin) of his mother. It is clear that such a decision could not be spontaneous especially since such kindred marriages could be concluded only with the permission of the Pope of Rome. And once again the Dominic Fermat's money solved this not simple problem.

Louise's father was an adviser to the Toulouse Parliament and being in the service of King Louis XIII, received a noble title, so Pierre had no problems with employment. But it would be a delusion to expect that also further everything will go on easily and smoothly. After the end of the study, marriage and the beginning of work, the reality seemed to Pierre as at all not so rosy. The gray days of the hustle and bustle of earning money for daily bread went day after day and did not leave any hopes to be engaged in science. And then it was still a very great good to have within the framework of lawyer activity the ability to support though not a luxurious, but still a well-off life in those difficult times for France.

A new danger for Pierre appeared unexpectedly. The next plague epidemic claimed the life of his father-in-law and this could have a very bad effect on his fate. However, by that time he had already managed to establish friendly relations with other senators what opened for him the way to parliament and as a result it made possible to turn the misfortune in his favor. With the help of a fair amount of money, he still managed to take the vacant position of an official in charge of receiving complaints in the cassation chamber of the Toulouse parliament.

The biographers of Pierre Fermat rate his career as simply brilliant, but at that they lose sight of one very significant detail. Exactly such a career tightly closes him all even the slightest opportunities to be engaged in science. They did not take into account the fact that there is a royal directive forbidding the posts of councilors of parliament for the people engaged in scientific research that may contradict the Holy Scriptures. But since Pierre became a senator, this will put a big fat cross on his dreams of being engaged in science on a professional basis. He will carry this cross for the rest of his life.

Moreover, as a Catholic he should not commit any mortal sin and is obliged to confess regularly once a year about the pardonable sins committed by him. As such a pardonable sin Pierre reports at confession about his moderate idleness after reading the books by Diophantus of Alexandria “Arithmetic” and “Tasks undertraining and pleasant, related to numbers”.

Pic. 6. Diophantus of Alexandria

The risk of falling into disfavor by such a sin fall was small because the book was published by Claude Gaspard Bachet de Méziriac a flawless in every respect a high-ranking linguist and future member of the French Academy established by Cardinal Richelieu in 1635. Here of course, there will be a question about the secret of confession. But if even in our time with respect to the Catholic Church this question looks very naïve, then what is to say about the times when the supreme executors of the royal power were cardinals. All priests were obliged to inform the authorities about what their parishioners live and especially officials in government posts. Information from the priests was also controlled, for which authorized inspectors were sent to the places.

Pic. 7. Bachet de Méziriac

It is understandable that Pierre could not expect anything good from meeting with such an inspector, but he had no choice and was ready to put up the complete impossibility of his dream. But then of course, he could not have known that he was destined to another fate and it was to decide at that very moment. It is even difficult to imagine his amazement when an arrived inspector turned out to be the priest Marin Mersenne… a passionate lover and connoisseur of mathematics!!!

Pic. 8. Marin Mersenne

Pierre took it as the supreme wonder bestowed on him from heaven by the Almighty. And how else could this be understood since Reverend Father Mersenne managed miraculously to organize for him the possibility of correspondence with René Descartes himself as well as with other elite representatives of the French creative aristocracy what about he could not ever to dream. Pierre went through the test brilliantly when he was able to solve several problems at the request of Mersenne and in particular quickly calculate some of the so-called perfect numbers moreover, also those that were previously unknown. Hardly anyone else could to solve or at least somehow cope this task.

Historians in their studies see only pure randomness in the coincidence of interest to the numbers of Mersenne and Fermat, and Mersenne himself in their presenting is a weirdo acting on his whim. However, in real history so does not happen and there should be a more reasonable explanation of events. In this sense, it would be much more logical to believe that Mersenne was no more than a performer of some instruction from above, and since he came from church nobility, only one person could give such an instruction to him — it was no one other as Cardinal Armand-Jean du Plessis Duc de Richelieu!

Pic. 9. René Descartes

This implies the activity of the association of learned nobility created by Mersenne, could not be just his initiative, but was sanctioned by the highest authorities of that time, otherwise this activity could not be deployed or it would curtail after the death of Mersenne in 1648. However, his brainchild continued to function for a long time and successfully until the creation of the French Academy of Sciences in 1666.

As for Pierre Fermat who became a Senator, he found himself in a difficult position. His abilities were now in demand, but he could develop them only at his own expense and without the right of publication because no one has repealed the royal prescription for restricting appointments to the posts of advisers to parliaments and he had no other means to earn a living. So, for his future opponents, he will appear as a recluse who does not want to share the secrets of his scientific discoveries. Even his friend Blaise Pascal in one of his letters sincerely wondered why he did not publish his works. To this Fermat also sincerely replied, he did not at all want his name to appear in print. Well, he really could not refer to the royal directive, which does not allow any scientific activity on the position he occupies.

Pic. 10. Blaise Pascal

For Fermat everything was happened so that he had no opportunity to solve this problem otherwise as by his direct participation in the preparation of the royal decree on the creation of the French Academy of Sciences. This is indicated by his correspondence with Mersenne and Pierre de Carcavy who was involved in the preparation of this decree. Fermat received a desired noble title only after 17 years of diligent service becoming in 1648 a member of Edicts House, which met regularly in the little town of Castres near Toulouse. But this promotion only increased his workload and further limited his opportunity for science activity.

But paradoxically in this life drama is distinctly seen a truly divine providence having lay a special mission to Senator Pierre de Fermat aimed at saving science from destruction. At that early age the science was still seemed as a beautiful tree, which by growing became more and more valuable and attractive. But with the development of science the features of perfection and harmony inherent in it, began to fade and the image of the beautiful creation of the mind more and more resembled a helpless little freak.

Pic. 11. Pierre de Carcavy

These first signs of trouble were noticed else by Fermat since controversies in his correspondence with colleagues appeared almost on empty place. It became clear that this tree has almost no roots. This means that science does not have a sufficiently strong foundation and for it there is a threat of the fate of the Pisa Tower. Then, in order for this magnificent building of science to serve its intended purpose, all creative forces will have to be used not for development, but for preventing its complete collapse.

For Fermat this theme was going past the limits of his physical possibilities and he considered it only from the point of view of generalizing methods for solving various arithmetic problems. It is so, because arithmetic is not some separate science, but the basis for all other sciences. If we have no arithmetic, then we have no any science generally. In this sense, the arithmetic tasks proposed by Fermat are of peculiar importance. Their peculiarity is that they teach people to think in general categories i.e. to find methods regulating the possibilities of computations for solving a wide range of tasks.

And here is an amazing paradox. About Diophantus who gave solutions to nearly two hundred completely not simple arithmetic tasks, now, if anyone remembers of him, then only in connection with the name of Fermat. But about Fermat himself, who did not leave any single (!!!) proof of his theorems,⁵ all and sundry are constantly discussing for the fourth century in a row! Very few of those who were able to solve although one Fermat’s tasks, secured for themselves world-wide fame, but countless number of people who suffered fiasco, cannot find for this any rational explanation and they have no other choice, but only simply to ignore this very fact.

But how could such an amazing phenomenon appear in the history of science when a man, who was not even a professional scientist, became so famous? To see here only an accidental combination of circumstances, would be clearly unwise. It is much more logical to proceed from the fact that at some stage in his life, Fermat began to realize that if his plans for publishing his research were carried out, the fate of Diophantus, which was already then almost forgotten, awaits him at best. If about Fermat anyone will also remember, then only against the background of derogatory and even caricature opinions of the “experts”.

In fact, it is all happened just so, but the effect was the opposite. No one could have imagined that thanks to Fermat a fascination with mathematics would take on such a mass character. The more his opponents sought to belittle him, the more popular his name became. Even the feats of D'Artagnan, which were fictional by A. Dumas, were simply childish pranks compared to what his fellow countryman Toulousean Senator Pierre de Fermat did in reality. And yet, how could this provincial judicial official be able to achieve such an amazing result?

It is very simple since he was a lawyer, he did everything exclusively and only legally, therefore he has left to himself all the works, in which his opponents could see recordings of “heretical content”. In addition, he was not only an outstanding mind with a lot of life experience, but also a Gascon. And it is well known that people of this type even the very serious doings can present in such an unpretentious and humorous wrapper: Yes, sometimes I'm reading Diophantus' “Arithmetic” at leisure and made notes with some ideas following the example of the esteemed and Right Honorable Claude Bachet who performed not only the Latin translation during the preparation of this book in 1621, but also added his own remarks.

Fermat did exactly the same i.e. had prepared for publication, as if were not his own works, but the same “Arithmetic” of Diophantus (see Pic. 96 in Appendix VI) with the same remarks of Bachet and has added to them the 48 his remarks. Everything was prepared so that any claims to this book or to him the Honorable Senator Pierre de Fermat simply could not be. But when the book was published, then unlike its previous editions, it stirred up the entire scientific world! Those comments made allegedly in passing on the margins of Diophantus’ book, turned out to be so valuable that they allowed scientists to develop science very noticeably using Fermat's new ideas for hundreds of years! And everything would be just perfect if it were not this his Last Theorem not amenable to any comprehension in scientific circles.

It would seem, what might be unusual here? Such unresolved problems in science are simply cannot be counted. But the fact of the matter is that the author of the theorem himself announced that he had the"truly amazing proof", but science cannot get although any for 350 years!!! It is only in the mass consciousness the author of the theorem is a real triumphant, but for science it’s like a bone in the throat. Here are already present obvious signs of illness. What kind of science is this, which for hundreds of years cannot to solve the school task? It would be OK if only one this task, but science cannot also recognize the obvious fact that it does not have the basic knowledge necessary for this, which Fermat discovered yet in those distant times.

Science lost not only the ability to comprehend, but also to orient in the events around it. How is it so that we have no knowledge, there are a whole mountain of them! This is for sure, “knowledge” was accumulated so many that to understand and assimilate all this wealth has become beyond human strength and capabilities. But in fact, everything is just the opposite. There is a very noticeable lack of real knowledge and the most part of all what has been accumulated, is empty grinding of many problems, which either have no solutions at all or else worse when dubious ideas are taken as the initial ones, on which mind-blowing theories are built, what naturally generating all sorts of paradoxes and contradictions. Then scientists are trying with all their might to overcome them, but for some reason if something works out for them that only with the help of even more mind-blowing theories.

Such an unusual character of our perceptions concerning to science, can cause a very negative reaction. But here we can confess that we had very good reasons for this because we managed to look in those very “heretical recordings” of Fermat. For greater persuasiveness we directly here will show one of the examples of our capabilities and accurately reproduce the real text of the most intriguing recording of the Fermat's Last theorem in the margins of Diophantus'"Arithmetic", which instance did belong to the author and disappeared unknown whither. So, in this place (see Pic. 5), we gain sight of several notes to the task under the number VIII made in Latin at different times. In translation they look like this:

1st entry: However, it is impossible to decompose C into two other C or QQ into two other QQ. Both proof by the descent method.

2nd entry: The second case is impossible because the number 2aabb is not a square.

3rd entry: New solution to the Pythagorean equation AB=2Q.

4th record: It may be computed as many numbers aa+bb–cc=a+b–c as you like.

5th entry: And in general, it is impossible to decompose any power greater than 2 into two powers with the same index. Proof by a key formula method.

6th entry: However, you can calculate as many numbers C+QQ=CQ as you like.

Now this restored text in margins of book can be compared with the text published in the edition"Arithmetic"by Diophantus with Fermat’s comments in 1670 (see Pic. 3 and at the end of Pt. 4.2):

However, it is impossible to decompose a cube into two cubes or a biquadrate into two biquadrates and generally any power greater than two, into two powers with the same index. I have discovered truly amazing proof of this, but these margins are too narrow to put it here.

But then it turns out the recovered text is not at all the same that was published. Well, of course not that one! It's clear, if you publish the real text of the remarks made in the margins of the book, then no one will understand anything because that who writes them, does it not for someone, but only for himself. On the other hand, it is obvious that the content of the recordings in the margins is so that they could not be made in the course of reading the book and are the result of a very voluminous and many years of work that was done separately. It is obvious that in addition to these short notes there is yet a whole bunch of papers in draft and finishing versions with brief or detailed explanations. These papers have not always been prepared for printing and they still need to be brought to the desired state. Hence it is clear why the text was edited accordingly for publication in 1670. From the real notes all was removed that reveals the method of proof and the sequence of solving individual tasks, which have eventually led to the discovery of the FLT.

The restored remarks follow in chronological order and may diverge in time over years. The margins' records of the book were made after they were prepared separately, but it was not intended that they be published in the same view. On the contrary, in the final formulation of the FLT everything that could be concealed from the history and components of this brilliant scientific discovery, was completely removed. Only the final result has been remained, which turned out to be beyond the powers of all subsequent science right up to the beginning of the XXI century!

If this reconstruction of the original FLT recording on the margins of the book appeared 30 years earlier, it would have caused a quite stir in the scientific world since the sixth entry develops (!!!) this theorem to the general case with the different of power's indexes! However, this stir did nevertheless take place 25 years ago, and again it was caused not by a professional, but by an amateur interested in FLT with his conjecture corresponding to the restored sixth entry. Of course, to believe in all this is not easily, but also to invent such a thing is also hardly possible. Now we have to explain in more detail these restored entries in the margins and this will be done in the next points of our work and the same senator who started this whole story, will help us in this.

Примечания

1

Naturalized geometric elements form either straight line segments of a certain length or geometric figures composed of them. To make of them figures with curvilinear contours (cone, ellipsoid, paraboloid, hyperboloid) is problematic, therefore it is necessary to switch to the representation of geometric figures by equations. To do this, they need to be placed in the coordinate system. Then the need for naturalized elements disappears and they are completely replaced by numbers for example, the equation of a straight line on the plane looks as y=ax+b, and the circle x²+y²=r², where x, y are variables, a, b are constants offset and slope straight line, r is the radius of the circle. Descartes and independently of him Fermat had developed the fundamentals of such (analytical) geometry, but Fermat went further proposing even more advanced methods for analyzing curves that formed the basis of the Leibniz — Newton differential and integral calculus.

2

Under conditions when the general state of science is not controlled in any way, naturally, the process of its littering and decomposition is going on. The quality of education is also uncontrollable since both parties are interested in this, the students who pay for it and the teachers who earn on it. All this comes out when the situation in society becomes conflict due to poor management of public institutions and it can only be “rectified” by wars and the destruction of the foundations of an intelligence civilization.

3

The name itself “the Basic theorem of arithmetic”, which not without reason, is also called the Fundamental theorem, would seem a must to attract special attention to it. However, this can be so only in real science, but in that, which we have, the situation is like in the Andersen tale when out of a large crowd of people surrounding the king, there is only one and that is a child who noticed that the king is naked!

4

On a preserved tombstone from the Fermat’s burial is written: “qui literarum politiforum plerumque linguarum” — skilled expert in many languages (see Pic. 93-94 in Appendix VI).

5

It is believed that Fermat left only one proof [36], but this is not entirely true since in reality it is just a verbal description of the descent method for a specific problem (see Appendix II).