The Wonders of Arithmetic from Pierre Simon de Fermat. Youri Veniaminovich Kraskov
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:
43
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.
44
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.
45
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.
46
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.
47
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.
48
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.
49
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.
50
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.
51
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 x2, then the 2nd will be 16-x2. 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 4x2–16 x+16. It should be 16-x2. Add the missing to both sides and subtract the similar ones from the similar ones. Then 5x2 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].
52
If c2= p2N2 and p2 (as well as any other pi2 of prime factors c) does not decomposed into a sum of two squares i.e. p2=q2+r where r is not a square then c2=p2(q2+r)=(pq)2+p2r and here in all variants of numbers q and r it turns out that p2r also is not a square then the number c2 also cannot be the sum of two squares.
53
This discovery was first stated in Fermat’s letter to Mersenne dated December 25, 1640. Here, in item 2-30 it is reported: “
54
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).
55
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: “