The Wonders of Arithmetic from Pierre Simon de Fermat. Youri Veniaminovich Kraskov

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.

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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).

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!

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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.

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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”.

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

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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