The Wonders of Arithmetic from Pierre Simon de Fermat

In this book there are examples of such calculations, which leave no room for doubts that it would be impossible to carry them out without knowing the essence … no, not of a conjecture, but of a much stronger statement called here “The Beal Theorem”! If the aim of the Beal Prize is really to get this impressive scientific discovery, then the organizing Committee in the face of "American Mathematical Society" would be easier do not rely on the propitiousness of mathematical editions, and just to request it directly from the author of this book.

This way would be clearly simpler and better since the proof of the Beal conjecture is too elementary and not so significant for science as the proof of the Beal Theorem, which would be much more useful, productive and impressive with the same end result that is required in conditions of the Beal Prize. The risk of arising another fake in this case will be excluded, but if nothing to be done to solve this problem, the initiator of the prize Mr. Andrew Beal may never wait to achieve his goal. Besides, it should be borne in mind that expert evaluation of the Beal conjecture proof does not require such obviously excessive precautions, because this task is for children from secondary school. What is written in this book is more than enough to make sure that this task has not any difficulties for the author.

It is very curious in this sense, how science will react to the appearance here of the FLT proof, performed by Fermat himself! And this is in conditions when as many as 18 (!!!) the most prestigious awards for obviously erroneous proof 1995 have already been presented! Of course, no one is immune from errors and we will show here how such pillars of science as Euclid and Gauss made the most elementary blunders in proving the Basic theorem of arithmetic, as well as Euler, who blessed the use in algebra of “complex numbers” , which are not numbers due to the fact that they do not obey to this same Basic theorem. However, Euler wasn’t aware of it yet, but his followers know this perfectly well for the two hundred years, nevertheless no one even had a finger stir to correct this mistake.

As for the not needed scientific discoveries, many people simply do not know that they can live quietly and consume all the vital resources they need only until the knowledge resource, accumulated in society for a given level of its development, will be exhausted. And after that, in order to keep what has been achieved, the stronger countries will attack the weaker ones and live at the expense of their plunder. But this would not have been necessary at all if these “strong” countries had enough knowledge. Then they would not have conflict with the rest of the world since all the necessary resources would be provided in abundance by science.

On this we will complete our introduction, but we will give it such a secret impulse that will allow us to perform a real wonder! … no, even two! We can call these wonders here by their proper names because our eternal opponents from the complete lack of real science by them, are simply incapable of this.

As a result, they will learn about the realization of the most grandiose technological breakthrough in Russia in the entire history of our civilization, with unlimited potential of development effectiveness for the immense future. The notorious “valleys”, “techno parks”, “incubators” and the like ghosts for such breakthroughs are unsuitable in principle. But still earlier, another wonder will happen when Russia literally in a couple of months, on the wreckage of collapsing today the world usury financial system, will create a new one, in which no any international money will be needed and all countries in international trade will use only their national currencies.

Are you again don't believe? Well, you can see for yourself because the book is in your hands!

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 - 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 x

+y

=r

, 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.]

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 - 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.]

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 - 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!]

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 Fran?oise 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 languages[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).] 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 Mеziriac 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 na?ve, 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 Mеziriac

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 Renе 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. Renе 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 - 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).] 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.