The Wonders of Arithmetic from Pierre Simon de Fermat

+ 217519

+ 414560

= 422481

. Another example 2682440

+15365639

+18796760

=20615673

. For the fifth power everything is much simpler. 27

+84

+110

+133

=144

. 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.]

Pic. 28. David Hilbert

Following the example of Euler on the eve of the 20th century, Hilbert offered to the scientific community 23 problems, which according to his assumption, are unlikely to be solved in the foreseeable future. Nevertheless, Hilbert's colleagues coped with them rather quickly, while Euler’s hypothesis has held almost until the 21st century and was only refuted with the help of computers, what is also described in Singh’s book. So, the suspicion that the FLT was merely an assumption of its author has lost any reason.

Hilbert had not cope with overcoming contradictions in set theory and could not do it because this problem is not at all mathematical, but informational one, so computer scientists should solve it sooner or later and when this happened, they are surprisingly very easily (and absolutely true) found a solution just forbidding closed chains of links.[23 - 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.] It is clear Hilbert could not know about it then and decided that the most reliable barrier to contradictions can be provided with the help of axioms. But axioms cannot be created on empty place and must come out of something and this something is a number, but what it is, no one can explain this not then nor now.

A brilliant example of what can be created with axioms is given in the same book of Singh. The obvious incident with the lack of a clear formulation to the notion of a number can accidentally spoil any rainbow picture and something needs to be done with it. It gets especially unpleasant with the justification of the “complex numbers”. Perhaps this caused the appearance in the Singh’s book of Appendix 8 called “Axioms of arithmetic”, in which 5 previously known axioms relating to a count are not mentioned at all (otherwise the idea will not past), while those that define the basic properties of numbers are complemented and a new axiom appears so that it must exist the numbers n and k, such that n+k=0 and then everything will be in the openwork!

Of course, Singh himself would never have guessed this. It is clearly visible here the help of consultants who for some reason forgot to change the name of the application since these are no longer axioms of arithmetic because already nothing is left of it.[24 - 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.] The school arithmetic, which for a long time barely kept on the multiplication table and the proportions, is now completely drained. Instead it, now there is full swing mastering of the calculator and computer. If such “progress” continues further, then the transition to life on trees for our civilization will occur very quickly and naturally.

Against this background a truly outstanding scientific discovery was made in Wikipedia, which simply has no equal in terms of art and the scale of misinformation. For a long time, many people thought that there are only four actions of arithmetic, these are addition and subtraction, multiplication and division. But no! There are also exponentiation and … root extraction (???). The authors of the articles given us this "knowledge" through Wikipedia clearly blundered because extracting the root is the same exponentiation only not with the integer power, but with fractional one. No of course, they knew about it, but what they didn’t guess was that it was they who copied this arithmetic action at Euler himself from that very book about the wonder-algebra[25 - 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.].

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

follows x=log

y. 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 - 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

+b

-c

(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 - 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 - 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 - 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

?1 by the number x?1, or the number x

?1 by

the number x±1, or the number x

+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

=151114054

+53969305

=82736654

+137487415

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 - To solve this problem, you need to use the formula that presented as the identity: (a

+b

)?(c

+d

)=(ac+bd)