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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" became an impossible task for our entire science. Then naturally, the question arises how did the authors of these books manage to do such creations? Of course, this question also arose in science, but instead of answering it only retains its proud silence. Well, then nothing prevents us from expressing our version here.
Apparently, there were somehow preserved and then restored written sources of knowledge from a highly developed civilization perished in earlier times. Only especially gifted people with extrasensory abilities allowing them to understand written sources regardless of the carrier and language, in which they were presented, could read and restore them. Euclid who was most likely a king, involved a whole team of such people, while Diophantus coped itself one and so the authorship of both appeared although in fact it was not the scientists who worked on the books, but only scribes and translators. But now we come back to the very task 8 from the second book of “Arithmetic” by Diophantus: Decompose a given square into the sum of two squares.
In the example of Diophantus, the number 16 is divided into the sum of two squares and his method gives one of the solutions 42=202/52=162/52+122/52 as well as countless other similar solutions51. However, this is not a solution to the task, but just a proof that any integer square can be made up of two squares any number of times either in integer or in fractional rational numbers. It follows that the practical value of the Diophantus method is paltry since from the point of view of arithmetic, the fractional squares are nonsense like, say, triangular rectangles or something like that. Obviously, this task should be solved only in integers, but Diophantus does not have such a solution and of course, Fermat seeks to solve this problem himself especially since at first, he sees it as not at all complicated.
So, let in the equation a2+b2=c2 given the number c and you need to find the numbers a and b. The easiest way to find a solution is by decomposing the number c into prime factors: c=pp1p2…pk; then
c2=p2p12p22…pk2=p2(p1p2…pk)2=pi2N2
Now it becomes obvious that the number c2 can be decomposed into a2 + b2 only if at least one of the numbers pi2 also decomposes into the sum of two squares.52 But this is a vicious circle because again you need to decompose square into a sum of two squares. However, the situation is already completely different because now you need to decompose a square of prime number and this circumstance becomes the basis for solving the task. If a solution is possible, then there must exist such prime numbers that decomposes into the sum of two squares and only in this case in accordance with the identity of the Pythagoreans, you can obtain:
pi2=(x2+y2)2=(x2−y2)2+(2xy)2
i.e. the square of such a prime will also be the sum of two squares. From here appears the truly grandiose scientific discovery of Fermat:53
All primes of type 4n+1 can be uniquely decomposed into the sum of two squares, i.e. the equation p=4n+1=x2+y2 has a unique solution in integers. But all other primes of type 4n−1 ca
In the Fermat's letter-testament it was shows how this can be proven by the descent method. However, Fermat’s proof was not preserved and Euler who solved this problem had to use for this all his intellectual power for whole seven years.54 Now already the solution to the Diophantine task seems obvious. If among the prime factors of number c there is not one related to the type 4n+1, then the number c2 ca
c2= N2pi2= (Nx)2+(Ny)2
where x= u2−v2; y=2uv; a=N(u2−v2); b=N2uv
The solution is obtained, however it clearly does not satisfy Fermat because in order to calculate the number N you need to decompose the number c into prime factors, but this task at all times was considered as one of the most difficult of all problems in arithmetic.55 Then you need to calculate the numbers x, y i.e. solve the problem of decomposing a prime of type 4n+1 into the sum of two squares. To solve this problem, Fermat worked almost until the end of his life.
It is quite natural that when there is a desire to simplify the solution of the Diophantine task, a new idea also arises of obtaining a general solution of the Pythagoras’ equation a2 + b2 = c2 in a way different from using the identity of Pythagoreans. As it often happens, a new idea suddenly arises after experienced strong shocks. Apparently, this happened during the plague epidemic of 1652 when Fermat managed to survive only by some miracle, but it was after that when he quite clearly imagined how to solve the Pythagoras’ equation in a new way.
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 ca
53
This discovery was first stated in Fermat’s letter to Merse
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: “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.