Youri Kraskov - The Wonders of Arithmetic from Pierre Simon de Fermat

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).