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

The correct name of the sixth action of arithmetic is logarithm i.e. calculating the power index (x) for a given power number (y) and basis of a power (z) i.e. from y=z xfollows x=log zy. As in the case with the name of the Singh’s book, this error is not at all accidental since no one really worked on logarithms as part of the arithmetic of integers. If this happens someday, then not earlier than in some five hundred years! But as for the action with power numbers, the situation here is not much better than with logarithms. If multiplication and dividing of power numbers as well as exponentiation a power number to a power, do not present any difficulties, but the addition of power numbers is still a dark forest even for professors.

The clarification in this matter begins with the FLT, which states that the sum of two power integer with the same power index greater than the second, cannot be an integer with the same power index. In this sense, this theorem is not at all any puzzle, but one of the basic propositions that unequivocally (!) regulates the addition of integer powers, therefore, it is of fundamental importance for science. 26 26 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). The fact that the FLT has not yet been proven, indicates only the state of current science, which is falling apart right before our eyes. Science cannot even imagine that if the proof from Fermat himself came to us, it would have been long ago taught in secondary school.

Many people of course, will perceive it as a fairy tale, but only the completely blind ones may not notice that behind all this absurd and awkward history with the FLT, clearly and openly ears of the unholy stick so out, that he was enough to deprive human civilization of access to Fermat's works on arithmetic, so it immediately turned out to be completely disoriented. Instead of developing science they began being vigorously to destroy it and even with very good intentions. But a special zeal in people appears when they have the material stimulus.

Pic. 29. Andrew Beal

Texas entrepreneur Andrew Beal 27 27 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). had proposed his conjecture, the proof of which allegedly could lead to a very simple proof of the FLT. Since for the solution of this problem it was proposed first $ 5 thousand, then $ 100 thousand, and from 2013 – a whole million, then naturally it appeared many willing people who began diligently this task to solve. However, in the conditions when arithmetic has long ceased to be the primary basic of all knowledge and still does not know, what is a number, everything turned upside down i.e. one amateur enthusiast was able to set on the ears the whole official science and so, that it had in fact already acknowledged the experience of Baron Munchhausen lifting himself up, taking himself by his collar, wherewith science did not even try at least to conceal its own insolvency (see pt. 4.5).

By working in the intense and tireless search for the FLT proof, it has never even occurred to anyone to search for Fermat’s manuscripts with layouts and calculations, without which he could not do 28 28 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. . However, again from Singh’s book we learn that such an idea came to Euler who asked his friend living in Lausanne (a city not far from Toulouse) to look for at least a little piece of paper with Fermat’s instructions to the FLT proof. But nothing was found, however, they were looking for what we do not really need! It was necessary to look for a cache!!!

Here is the new puzzle, which is not easier! What else kind of cache? … Oh yes! The fact is that only those Fermat's works remained, which he itself had already prepared for publication since otherwise they would hardly have been published. But all the working manuscripts for some reason has disappeared. It looks very strange and it is possible that they can still be kept in the cache, which Fermat has equipped to store the material evidence necessary for him to work as a senator and high-ranking judge. It was quite reasonable to keep calculations and proofs there, since Fermat’s scientific achievements could significantly damage his main work if they were made public before the establishment of the French Academy of Sciences. 29 29 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.

If we could somehow look into this cache, what will we see there? To begin with, let's try to find some simple tasks there. For example, the one that Fermat could offer today for secondary school students:

Divide the number x n−1 by the number x−1 , or the number x 2n−1 by

the number x±1 , or the number x 2n+1+1 by the number x+1.

It is obvious that students with the knowledge of solving such a task will be simply a head over the current students who are trained in the methods of determining the divisibility by only some small numbers. But if they else know a couple of the Fermat's theorems, they can easily solve also the more difficult problem:

Find two pairs of squares, each of which adds up to the same number

in the seventh power, for example,

221 7=151114054 2+53969305 2=82736654 2+137487415 2

Compared to the previous task where calculations are not needed at all, in solving this task, even with a computer calculator you have to tinker with half an hour to achieve a result, while apart from understanding the essence of the problem solution, you need to show a fair amount of patience, perseverance and attention. And who understands the essence of the solution, will be able to find other solutions to this problem. 30

Of course, such tasks can cause a real shock to today's students and especially to their parents who will even demand not to “dry the brains” of children. But if children's brains are not filled with elementary knowledge and not trained by solving the corresponding tasks, they will wither by themselves. This is proven by the statistics of the steady decline in today's students IQ compared with their predecessors. Really in fact, the above tasks are only a warm-up for the young generation, but children could produce a real furor for mathematicians offering them some simple Fermat's theorems about magic numbers (see Pt. 4.4.). And this is else a big question, could these theorems being solved by today's professors or will they again need some three hundred years and the story with the FLT will repeat? However, the chances of them in contrast to previous times, are very high because magic numbers are a direct consequence of the same “truly amazing” proof of the FLT, about the existence of which we have direct written evidence from Fermat himself.