The Wonders of Arithmetic from Pierre Simon de Fermat. Youri Veniaminovich Kraskov
with Fermat’s remarks (see Pic. 3 and Pic. 96 from Appendix VI). And since then i.e. for three and a half centuries, science cannot cope with this task. Moreover, perhaps this is namely why the FLT became the object of unprecedented falsification in the history of mathematics. It is very easily to verify this since the main arguments of the FLT “proof” 1995 are well known and look as follows.
If the FLT were wrong, then there would be exist an elliptical “Frey curve” (???): y2=x(x−an)(x+bn) where an+bn=cn. But Kenneth Ribet has proven that such a curve cannot be modular. Therefore, it suffices to obtain a proof of the Taniyama – Shimura conjecture, that all elliptic curves must be modular, so that it simultaneously becomes a proof of the FLT. The proof was presented in 1995 by Andrew Wiles who became the first scientist that allegedly has proven the FLT.
However, it turns out that the “Frey curve" and together with it the works of Ribet and Wiles have with the FLT nothing to do at all!!!44 And as regards the “proof” of A. Wiles the conjecture of Taniyama – Shimura, he also himself admitted45 that one needs much more to learn (naturally, from Wiles) in order to understand all of its nuances, setting forth on 130 pages (!!!) of scientific journal "Annals of Mathematics". Quite naturally that after the appearance of such exotic “proof”, scientists cannot come to their senses from such a mockery of science, the Internet is replete with all sorts of refutations,46 and there is no doubt that any generally accepted proof of the FLT still does not exist.
The special significance of the FLT is that in essence, this is one of the simple cases to addition of power numbers when only the sum of two squares can be a square and for higher powers such addition is impossible. However, according to the Waring-Hilbert theorem, any natural number (including an integer power) can be the sum of the same (or equal to a given) powers47. And this a much more complex and no less fundamental theorem was proven much earlier than the FLT.
We also note the fact that the FLT attracts special attention not at all because this task is simple in appearance, but very difficult to solve. There are also much simpler-looking tasks, which are not only not to be solved, but also even nobody really knows how to approach them 48. The FLT especially differs from other tasks that attempts to find its solution lead to the rapid growth of new ideas, which become impulses for the development of science. However, there was so much heaped up on this path that even in very voluminous studies, all this cannot be systematized and combined.49
Great scholars did not attach much importance to building the foundations of science apparently considering such creativity to be a purely formal matter, but centuries-old failures with the FLT proof indicate that they underestimated the significance of such studies. Now when it became clear where such an effective scientific tool as the descent method could come from, as well as other tools based on understanding the essence of number, it becomes clear why Fermat was so clearly superior to other mathematicians in arithmetic, while his opponents have long been in complete bewilderment from this obvious fact.
Here we come to the fact that the main reason for failures in the search for FLT proof lies in the difference between approaches to solving tasks by Fermat and other scientists, as well as in the fact that even modern science has not reached the knowledge that already was used by Fermat in those far times. This situation needs to be corrected because otherwise the FLT so will continue to discredit whole science.
One of the main questions in the studies on the FLT was the question of what method did Fermat use to prove this theorem? Opinions were very different and most often it was assumed that this was the method of descent, but then Fermat himself hardly called it "truly amazing proof." He also could not apply the Kummer method, from which the best result was obtained in proving the FLT proof over the last 170 years. But perhaps he besides the descent method had also other ones? Yes indeed, this is also described in detail in treatise "A New Discovery in the Art of Analysis" by Jacques de Billy [36]. There, he sets out in detail Fermat's methods, which allow him to find as many solutions as necessary in systems of two, three, or more equations. But here his predecessors Diophantus, Bachet and Viet at best found only one solution. After demonstrating Fermat's methods for solving the double equalities Billy also points to the most important conclusion, which follows from this: This kind of actions serves not only to solve double equalities, but also for any other equations.
Now it remains only to find out how to use the system of two equations to prove the FLT? Obviously, mathematicians simply did not pay attention to such an explicit clue from Fermat or did not understand its meaning. But for us this is not a problem because we can look into the cache and delve into the "heretical writings"! Based on what we have already been able to recover from Fermat’s works, we can now begin to uncover this greatest mystery of science, indicating also an effective method that allows us to solve the problem of FLT proof.
How it wouldn't be surprising, the essence of this method is quite simple. In the case when there are as many equations as there are unknowns in them, such a system is solved by ordinary substitutions. But if there is only one equation with several unknowns, then it can be very difficult to establish whether it can even have any solutions in integers. In this case, the numbers supposed as solutions can be expressed in the form of another equation called the “Key Formula” and then the result can be obtained by solving a system of two equations. Similar techniques when some numbers are expressed through others, have always been used by mathematicians, but the essence of the key formula is in another, it forms exactly that number, which reflects the essence of the problem and this greatly simplifies the way to solving the original equation. In such approaches and methods, based on an understanding the essence of numbers, in fact also lies the main superiority of Fermat over other scientists.50
To make it possible to follow the path that Fermat once laid, you need to find the starting element from the chain of events leading to the appearance of the FLT, otherwise there will be very little chance of success because everything else is already studied far and wide. And if we ask the question exactly this way, we will suddenly find that this very initial element was still in sight from 1670, but since then no one has paid any attention to it at all. However, in fact, we are talking about the very problem under number 8 from the book II of Arithmetic by Diophantus, to which also Fermat’s remark was written, became later a famous scientific problem. Everyone thought that this simple-looking problem has no difficulties for science and only Fermat had another opinion and worked for many years to solve it. As a result, he not only obtained it, but in addition to this he secured to his name unfading world fame.
4.1.2. Diophantus' Task
The book entitled “Arithmetic” by Diophantus is very old however, probably it appeared not in III as it was thought until recently, but in the XIV or XV century. In those times when yet there were no print editions, it was a very impressive in volume manuscript consisting of 13 books, from which only six ones reached us. In today's printed form this is a small enough book with a volume of just over 300 pages [2, 27].
In France an original Greek version of this book was published in 1621 with a Latin translation and comments from the publisher, which was Bachet de Méziriac. This publication became the basis for Fermat's work on arithmetic. The contents of the book are 189 tasks and solutions are given for all them. Among them are both fairly simple and very difficult tasks. However, since they have been solved, a false impression is created that these tasks are not educational, but rather entertaining ones i.e., they are needed not to shape science, but to test for quick wit. In those times, it could not have been otherwise because even just literate people who could read and write were very rare.
However, from the point of view the scientific significance of the presented tasks and their solutions, the creation of such a book is not something that to the medieval Diophantus, but to all scientists in the entire visible history would be absolutely impossible. Moreover, even at least properly understanding the contents of the Euclid's "Elements" and the Diophantus' "Arithmetic"
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 an+bn=cn 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 an+bn=cn i.e. Frey's curve y2=x(x−an)(x+bn) 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 an+bn=cn 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 32+42=52 Euler found a very simple and beautiful example of adding powers: 33+43+53=63. 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.