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^>2+y^>2=r^>2, 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.

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.

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!

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).

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).

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!

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.

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.

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.

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”.

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

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.

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

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

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

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

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

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

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

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

Waring's problem is the statement that any positive integer N can be represented as a sum of the same powers x_>i^>n, i.e. in the form N = x_>1^>n + x_>2^>n + … + x_>k^>n. It was in very complex way first proven by Hilbert in 1909, and in 1920 the mathematicians Hardy and Littlewood simplified the proof, but their methods were not yet elementary. And only in 1942 the Soviet mathematician Yu. V. Linnik has published arithmetic proof using the Shnielerman method. The Waring-Hilbert theorem is of fundamental importance from the point of view the addition of powers and does not contradict to FLT since there are no restrictions on the number of summands.

A counterexample refuting Euler’s hypothesis is 95800^>4 + 217519^>4 + 414560^>4 = 422481^>4. Another example 2682440^>4+15365639^>4+18796760^>4=20615673^>4. For the fifth power everything is much simpler. 27^>5+84^>5+110^>5+133^>5=144^>5. It is also possible that a general method of such calculations can be developed if we can obtain the corresponding constructive proof of the Waring's problem.

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

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

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

This is evident at least from the fact, in what a powerful impetus for the development of science were embodied countless attempts to prove the FLT. In addition, the FLT proof, obtained by Fermat, opens the way to solving the Pythagorean equation in a new way (see pt. 4.3) and magic numbers like a+b-c=a^>2+b^>2-c^>2 (see pt. 4.4).

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

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

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

To solve this problem, you need to use the formula that presented as the identity: (a^>2+b^>2)×(c^>2+d^>2)=(ac+bd)^>2+(ad−bc)^>2=(ac−bd)^>2+(ad+bc)^>2. We take two numbers 4 + 9 = 13 and 1 + 16 = 17. Their product will be 13×17 = 221 = (4 + 9) × (1+16) = (2×1 + 3×4)^>2 + (2×4 − 3×1)^>2 = (2×1 − 3×4)^>2 + (2×4 + 3×1)^>2 = 14^>2 + 5^>2 = 10^>2 + 11^>2; Now if 221^>6 = (221^>3)^>2 = 10793861^>2; then the required result will be 221^>7 = (14^>2 + 5^>2)×10793861^>2 = (14×10793861)^>2 + (5×10793861)^>2 = 151114054^>2 + 53969305^>2 = (10^>2 + 11^>2)×10793861^>2=(10×10793861)^>2 + (11×10793861)^>2=107938610^>2 + 118732471^>2; But you can go also the other way if you submit the initial numbers for example, as follows: 221^>2 = (14^>2 + 5^>2)×(10^>2 + 11^>2) = (14×10 + 5×11)^>2 + (14×11 − 5×10)^>2 = (14×10 − 5×11)^>2 + (14×11+5×10)^>2 = 195^>2 + 104^>2 = 85^>2 + 204^>2; 221^>3 = 221^>2×221 = (195^>2 + 104^>2)×(10^>2 + 11^>2) = (195×10 + 104×11)^>2 + (195×11 − 104×10)^>2 = (195×10 − 104×11)^>2 +(195×11 + 104 × 10)^>2 = 3 094^>2 + 1105^>2 = 806^>2 + 3185^>2; 221^>4 = (195^>2 + 104^>2)×(85^>2 + 204^>2) = (195×85 + 104×204)^>2 + (195×204 − 85×104)^>2 = (195×85 − 104×204)^>2 + (195×204 + 85×104)^>2 = 37791^>2 + 30940^>2 = 4641^>2 + 48620^>2; 221^>7 = 221^>3×221^>4 = (3094^>2 + 1105^>2)×(37791^>2 + 30940^>2) = (3094×37791 + 1105×30940)^>2 + (3094×30940 − 1105×37791)^>2 = (3094×37791 − 1105×30940)^>2 + (3094×30940 + 1105×37791)^>2; 221^>7 = 151114054^>2 + 53969305^>2 = 82736654^>2 + 137487415^>2

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

For mathematicians and programmers, the notion of function argument is quite common and has long been generally accepted. In particular, f (x, y, z) denotes a function with variable arguments x, y, z. The definition of the essence of a number through the notion of function arguments makes it very simple, understandable and effective since everything what is known about the number, comes from here and all what this definition does not correspond, should be questioned. This is not just the necessary caution, but also an effective way to test the strength of all kinds of structures, which quietly replace the essence of the number with dubious innovations that make science gormlessly and unsuitable for learning.

An exact definition the notion of data does not exist unless it includes a description from the explanatory dictionary. From here follows the uncertainty of its derivative notions such as data format, data processing, data operations etc. Such vague terminology generates a formulaic thinking, indicating that the mind does not develop, but becomes dull and by reaching in this mishmash of empty words critical point, it simply ceases to think. In this work, a definition the notion of “data” is given in Pt. 5.3.2. But for this it is necessary to give the most general definition the notion of information, which in its difficulty will be else greater than the definition the notion of number since the number itself is an information. The advances in this matter are so significant that after they will follow a real technological breakthrough with such potential of efficiency, which will be incomparably higher than which was due to the advent of computers.

Computations are not only actions with numbers, but also the application of methods to achieve the final result. Even a machine can cope with actions if the mind equips it with appropriate methods. But if the mind itself becomes like a machine i.e. not aware the methods of calculation, then it is able to create only monsters that will destroy also him selves. Namely to that all is going now because of the complete lack of a solution to the problem of ensuring data security. But the whole problem is that informatics as a science simply does not exist.

Specialists who comment on the ancients in their opinion the Euclid's "Elements" and the Diophantus' "Arithmetic", as if spellbound, see but cannot acknowledge the obvious. Neither Euclid nor Diophantus can be the creators the content of these books, this is beyond the power of even modern science. Moreover, these books appeared only in the late Middle Ages when the necessary writing was already developed. The authors of these books were just translators of truly ancient sources belonging to another civilization. Nowadays, people with such abilities are called medium.

If from the very beginning we have not decided on the concept of a number and have an idea of it only through prototypes (the number of fingers, or days of the week etc.), then sooner or later we will find that we don’t know anything about numbers and follow the calculations an immense set of empirical methods and rules. However, if initially we have an exact definition the notion of number, then for any calculations, we can use only this definition and the relatively small list of rules following from it. If we ourselves creating the required numbers, we can do this through the function arguments, which are represented in the generally accepted number system. But when it is necessary to calculate unknown numbers corresponding to a given function and task conditions, then special methods will often be required, which without understanding the essence of numbers will be very difficult.

The content of Peano’s axioms is as follows: (A1) 1 is a natural number; (A2) For any natural number n there is a natural number denoted by n' and called the number following n; (A3) If m'=n' for any positive integers m, n then m = n; (A4) The number 1 does not follow any natural number i.e. n' is never equal to 1; (A5) If the number 1 has some property P and for any number n with the property P the next number n' also has the property P then any natural number has the property P.

In the Euclid's "Elements" there is something similar to this axiom: "1. An unit is that by virtue of each of the things that exist is called one. 2. A number is a multitude composed of units” (Book VII, Definitions).

So, count is the nominate starting numbers in a finished (counted) form so that on their basis it becomes possible using a similar method to name any other numbers. All this of course, is not at all difficult, but why is it not taught at school and simply forced to memorize everything without explanation? The answer is very simple – because science simply does not know what a number is, but in any way cannot acknowledge this.

The axioms of actions were not separately singled out and are a direct consequence of determining the essence a notion of number. They contribute both to learning and establish a certain responsibility for the validity of any scientific research in the field of numbers. In this sense, the last 6th axiom looks even too categorical. But without this kind of restriction any gibberish can be dragged into the knowledge system and then called it a “breakthrough in science”.

The reconstructed proof of Fermat excludes the mistake made by Euclid. However, beginning from Gauss, other well-known proofs the Basic theorem of arithmetic repeat this same mistake. An exception is the proof received by the German mathematician Ernst Zermelo, see Appendix I.

Facsimile of the edition with the Cauchy's GTF proof was published by Google under the title MEMIRES DE LA CLASSE DES SCIENCES MATHTÉMATIQUES ET PHYSIQUES DE L’INSTITUT DE France. ANNEES 1813, 1814, 1815: https://books.google.de/books?id=k2pFAAAAcAAJ&pg=PA177#v=onepage&q&f=false What we need is on page 177 under the title DEMONSTRATION DU THÉORÉME GÉNÉRAL DE FERMAT, SUR LES NOMBRES POLYGONES. Par M. A. L. CAUCHY. Lu à l’Académie, le 13 novembre 1815 (see Pics 34, 35). The general proof of Cauchy takes 43 (!!!) pages and this circumstance alone indicates that it does not fit into any textbook. Such creations are not something that students, but also academics are not be available because the first cannot understand anything about them and the second simply do not have the necessary time for this. Then it turns out that such proofs are hardly possible to check how convincing they are i.e. are they any proofs in general? But if Cauchy applied the descent method recommended by Fermat, then the proof would become so convincing that no checks would be required. A very simple conclusion follows from this: The Fermat's Golden Theorem as well as some of his other theorems, still remain unproven.

Examples are in many videos from the Internet. However, these examples in no way detract from the merits of professors who know their job perfectly.

It must be admitted that the method of Frey's proof is basically the same as that of Fermat i.e. it is based on obtaining a solution to the equation a^>n+b^>n=c^>n by combining it into a system with another equation – a key formula, and then solving this system. But if Fermat’s key formula a+b=c+2m is derived directly from the initial equation, while at Frey it is just taken from nothing and united to the Fermat equation a^>n+b^>n=c^>n i.e. Frey's curve y^>2=x(x−a^>n)(x+b^>n) is a magical trick that allows to hide the essence of the problem and replace it with some kind of illusion. Even if Frey could prove the absence of integer solutions in his equation then this could in no way lead him to the proof of the FLT. But he did not succeed it also, therefore one “brilliant idea” gave birth to an “even more brilliant idea” about the contradiction of the “Frey curve” to the Taniyama – Simura conjecture. With this approach you can get incredibly great opportunities for manipulating and juggling the desired result, for example, you can "prove" that the equation a+b+c=d as well as the Fermat equation a^>n+b^>n=c^>n in integers cannot be solved if take abc=d as a key formula. However, such "ideas" that obviously indicate the substitution the subject of the proof should not be considered at all, since magicians hope only for the difficulty of directly refuting their trick.

Here is how E. Wiles himself comments on a mistake found in his “proof” in 1993: “Even explaining it to a mathematician would require the mathematician to spend two or three months studying that part of the manuscript in great detail”. See Nova Internet Publishing http://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html It turns out that this "proof" understood only by its author, while everyone else needs to learn and learn.

Such debunks are very detailed, but too redundant since the arguments of the main authors of the FLT “proof” by G. Frey and E. Wiles look so ridiculous that otherwise as by the hypnotic influence of the unholy it would impossible to explain why many years after 1995 for some reason none of the recognized pundits so have ever noticed that instead of FLT proof we have got a something completely different.

Similarly, to the example from Pythagoras 3^>2+4^>2=5^>2 Euler found a very simple and beautiful example of adding powers: 3^>3+4^>3+5^>3=6^>3. For other examples, see comment 22 in Pt. 2.

For example, the task of the infinity of the set of pairs of twin primes or the Goldbach task of representing any even natural number as the sum of two primes. And also, the solution to the coolest problem of arithmetic about an effective way to calculate prime numbers is still very far from perfect despite the tons of paper spent on research on this problem.

In particular, Edwards in his very voluminous book [6, 38], was not aware of the fact that Gauss solved the Fermat's task of decomposing a prime number type 4n + 1 into a sum of two squares. But it was this task that became a kind of bridge to the subsequent discovering the FLT. Fermat himself first reported it in a letter to Blaise Pascal on 09/25/1654 and this is one of the evidences that of all his scientific works, the FLT is really his last and greatest discovery.

The main and fundamental difference between Fermat's methods and the ones of other scientists is that his methods are universal enough for a very wide range of problems and are not directly related to a specific task. As a rule, attempts to solve a problem begin with trial calculations and enumeration of all possible options and those who think faster get correspondingly more opportunities to solve it. Fermat has another approach. He makes trying only for the purpose of bringing them to some universal method suitable for the given task. And as soon as it him succeeds, the task is practically solved and the result is guaranteed even if there is still a very large amount of routine calculations ahead. See for example, comment 30 in Pt. 2.

The original solution to the Diophantus' task is as follows. “Let it be necessary to decompose the number 16 into two squares. Suppose that the 1st is x^>2, then the 2nd will be 16-x^>2. I make a square of a certain number x minus as many units as there are in site of 16; let it be 2x-4. Then this square itself is 4x^>2–16 x+16. It should be 16-x^>2. Add the missing to both sides and subtract the similar ones from the similar ones. Then 5x^>2 is equal to 16x and x will be equal to 16 fifths. One square is 256/25 and the other is 144/25; both folded give 400/25 or 16 and each will be a square” [2, 27].

If c^>2= p^>2N^>2 and p^>2 (as well as any other p_>i^>2 of prime factors c) does not decomposed into a sum of two squares i.e. p^>2=q^>2+r where r is not a square then c^>2=p^>2(q^>2+r)=(pq)^>2+p^>2r and here in all variants of numbers q and r it turns out that p^>2r also is not a square then the number c^>2 also cannot be the sum of two squares.

This discovery was first stated in Fermat’s letter to Mersenne dated December 25, 1640. Here, in item 2-30 it is reported: “This number (a prime of type 4n+1) being the hypotenuse of one right triangle, its square will be the hypotenuse of two, cube – of three, biquadrate – of four etc. to infinity". This is an inattention that is amazing and completely unusual for Fermat, because the correct statement is given in the neighbor item 2-20. The same is repeated in Fermat’s remark on Bachet’s commentary to task 22 book III of Arithmetic by Diophantus. But here immediately after this obviously erroneous statement the correct one follows: “This a prime number and its square can be divided into two squares in only one way; its cube and biquadrate only two; its quadrate-cube and cube-cube only three, etc. to infinity" (see Pt. 3.6). In this letter Fermat apparently felt that something was wrong here, therefore he added the following phrase: “I am writing to you in such a hurry that I do not pay attention to the fact that there are errors and omit a lot of things, about which I tell you in detail another time”. This of course, is not that mistake, which could have serious consequences, but the fact is that this blunder has been published in the print media and Internet for the fourth century in a row! It turns out that the countless publications of Fermat's works no one had ever carefully read, otherwise one else his task would have appeared, which obviously would have no solution.

Euler's proof is not constructive i.e. it does not provide a method for calculating the two squares that make up a prime of type 4n+1 (see Appendix III). So far, this problem has only a Gauss' solution, but it was obtained in the framework of a very complicated system “Arithmetic of Deductions”. The solution Fermat reported is still unknown. However, see comment 172 in Appendix IV (Year 1680).

Methods of calculating prime numbers have been the subject of searches since ancient times. The most famous method was called the "Eratosthenes’ Sieve". Many other methods have also been developed, but they are not widely used. A fragment of Fermat’s letter with a description of the method he created, has been preserved the letter LVII 1643 [36]. In item 7 of the letter-testament he notes: “I confess that my invention to establish whether a given number is prime or not, is imperfect. But I have many ways and methods in order to reduce the number of divides and significantly reduce them facilitating usual work." See also Pt. 5.1 with comments 73-74.

Fermat discovered formula (2) after transforming the Pythagoras’ equation into an algebraic quadratic equation – see Appendix IV story Year 1652. However, an algebraic solution does not give an understanding the essence of the resulting formula. This method was first published in 2008 [30].

For example, if m = p_>1p_>2 then in addition to the first three solutions there will be others: A_>4=p_>1; B_>4=2p_>1p_>2^>2; A_>5=p_>2; B_>5=2p_>1^>2p_>2; A_>6=2p_>1; B_>6=p_>1p_>2^>2; A_>7=2p_>2; B_>7=p_>2p_>1^>2; A_>8=p_>1^>2; B_>8=2p_>2^>2; A_>9=p_>2^>2; B_>9=2p_>1^>2

Formula (7) is called Fermat Binomial. It is curious that the same name appeared in 1984 in the novel "Sharper than the epee" by the Soviet science fiction writer Alexander Kazantsev. This formula is not an identity because in contrast to the identity of Newton Binomial in addition to summands, there is also a sum of them, but with the help of Fermat Binominal it is easy to derive many useful identities in particular, factorization of the sum and difference of two identical powers [30], see also Pt. 4.4.

In this case, identity (9) indicates that the same key formula is substituted into the transformed key formula (2) or that the equation (8) we obtained, is a key formula (2) in power n. But you can go the reverse way just give the identity (9) and then divide into factor the differences of powers and such a way you can obtain (8) without using the Fermat Binominal (7). But this way can be a trick to hide the understanding of the essence because when some identity falls from the sky, it may seem that there is nothing to object. However, if you memorize only this path, there is a risk of exposure in a misunderstanding of the essence because the question how to obtain this identity, may go unanswered.

Taking into account that c−a=b−2m the expression in square brackets of equation (8) can be transformed as follows: (c++b)_>n − (a++2m)_>n = с^>n-1− a^>n-1+ c^>n-2b− a^>n-22m+ c^>n-3b^>2− a^>n-3(2m)^>2+ … +b^>n-1 − (2m)^>n-1; с^>n-1 − a^>n-1 = (с − a)(c++a)_>n-1; c^>n-2b − a^>n-22m = 2m(c^>n-2 − a^>n-2) + c^>n-2(b − 2m) = (c − a)[2m(c++a)_>n-2 + c^>n-2]; c^>n-3b^>2 −a^>n-3(2m)^>2 = (2m)^>2(c^>n-3 − a^>n-3) + c^>n-3(b^>2 − 4m^>2) = (c − a)[4m^>2(c++a)_>n-3 + c^>n-3(b +2m)]; b^>n-1 − (2m)^>n-1 = (b − 2m)(b++2m)_>n-1 = (c − a)(b++2m) _>n-1 All differences of numbers except the first and last, can be set in general form: c^>xb^>y − a^>x(2m)^>y=(2m)^>y(c^>x − a^>x) + c^>x[b^>y − (2m)^>y] = (c − a)(c++a)_>x(2m)^>y + (b − 2m)(b++2m)_>yc^>x = (c − a)[(c++a)_>x(2m)^>y + (b++2m)_>yc^>x] And from here it is already become clear how the number (c − a) is take out of brackets. Similarly, you can take out of brackets the factor a + b = c + 2m. But this is possible only for odd powers n. In this case, equation (10) will have the form A_>iB_>iC_>iD_>i = (2m)^>n, where A_>i = c – b = a − 2m; B_>i = c – a = b − 2m; C_>i = a + b = c + 2m; D_>i – polynomial of power n − 3 [30].

Equation (10) can exist only if (1) holds i.e. {a^>n+b^>n−c^>n}=0 therefore, any option with no solutions leads to the disappearance of this ghost equation. And in particular, there is no “refutation” that it is wrong to seek a solution for any combination of factors, since A_>iB_>i=2m^>2 may contradict E_>i=2^>n-1m^>n-2, when equating E_>i to an integer does not always give integer solutions because a polynomial of power n−2 (remaining after take out the factor c−a) in this case may not consist only of integers. However, this argument does not refute the conclusion made, but rather strengthens it with another contradiction because E_>i consists of the same numbers (a, b, c, m) as A_>i, B_>i where there can be only integers.

In this proof, it was quite logical to indicate such a combination of factors in equation (10), from which the Pythagoras’ numbers follow. However, there are many other possibilities to get the same conclusion from this equation. For example, in [30] a whole ten different options are given and if desired, you can find even more. It is easy to show that Fermat's equation (1) is also impossible for fractional rational numbers since in this case, they can be led to a common denominator, which can then be reduced. Then we get the case of solving the Fermat equation in integers, but it has already been proven that this is impossible. In this proof of the FLT new discoveries are used, which are not known to current science: there are the key formula (2), a new way to solve the Pythagoras’ equation (4), (5), (6), and the Fermat Binomial formula (7) … yes of course, else also magic numbers from Pt. 4.3!!!