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The Wonders of Arithmetic from Pierre Simon de Fermat. Youri Veniaminovich Kraskov
Читать онлайн.Название The Wonders of Arithmetic from Pierre Simon de Fermat
Год выпуска 2021
isbn 978-5-532-98629-9
Автор произведения Youri Veniaminovich Kraskov
Жанр Прочая образовательная литература
Издательство ЛитРес: Самиздат
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 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 xn−1 by the number x−1, or the number x2n−1 by
the number x±1, or the number x2n+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,
2217=1511140542+539693052=82736654 2+1374874152
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.
Reconstruction of this proof was briefly published as early as 2008 [30], but the unholy was on the alert and presented this event so, that modern science disoriented by the false notion that the problem was solved long ago, has not paid on this any attention. However, all secret sooner or later becomes clear and the decisive word in spite of everything, still remains for science. The question now is only when this science will finally awaken and comes to his senses. The longer it will be in a blissful state of oblivion, the sooner the terrible events will come that already now beginning to shake our world like never before.
In order for science to win a well-deserved victory over the gloom of ignorance and mass disinformation, which are triumphant today, it needs very little. For the beginning it is necessary simply to search for the very cache, in which such secrets of science are hidden, that have not lost their relevance for three and a half centuries.31 Even if the papers found in the cache will be unreadable, the very fact of the existence of the cache will be evidence that science is moving in the right direction and the results will not be long in coming.
We already did something in this direction when we restored the FLT recording in the margins of Diophantus 'Arithmetic' (see pic. 5 and the translation in the end of Pt. 1). Now, by all means, we need to get a complete picture of the whole sequence of events that led to the discovery of the FLT in its final wording published in 1670. It will not be easily at all, but since we got involved in this story, now we have nowhere to retreat and we will strain all our forces to achieve the aim. Fortunately, for this we have all the opportunities granted to us from above to get the coveted access to the cache of the Toulousean senator.
3. What is a Number?
3.1. Definition the Notion of Number
The question about the essence the notion of number at all times was for scientists the thing-in-itself. They of course, understood that they could not distinctly answer this question as well as they could not admit in this since this would have a bad effect on maintaining the prestige of science. What is the problem here? The fact is that in all cases a number must be obtained from other numbers, otherwise it cannot be perceived as a number. To understand for example, the number 365, you need to add three hundred with six tens and five units. It follows that the notion of a number does not decompose into components that are qualitatively different from it and in such a way as usual for science i.e. through analysis, it is not possible to penetrate the secret of its essence.
Scientists having a question about the nature of numbers immediately ran into this problem and came to the conclusion that a general definition the notion of number simply does not exist. But not a such was Pierre Fermat who approached this problem from other side. He asked: “Where does the notion of number come from?” And came to the conclusion that his predecessors were the notions “more”, “less” and “equal” as the comparisons’ results of some properties inherent to different objects [30].
If different objects are compared in some property with the same object then such a notion as a measurement appears, so perhaps is the essence of a number possible revealed through a measurement? However, it is not so. In relation to the measurement, the number is primary i.e. if there are no numbers, there can be no also measurements. Understanding the essence of the number becomes possible only after establishing the number is inextricably connect with the notion of “function”.
But this notion is not difficult to determine:
A function is a given sequence of actions with its arguments.
In turn, actions cannot exist on their own i.e. in the composition of the function in addition to them must include the components, with which these actions are performed. These components are called function arguments. From here follows a general definition the notion of number:
Number is an objective reality existing as a countable quantity, which consists of function arguments and actions between them.
For example, a+b+c=d where a, b, c are arguments, d is a countable quantity or the number value.32
To understand what a gap separates Pierre Fermat from the rest of the science’s world, it is enough to compare this simple definition with the understanding existing in today's science [13, 29]. But understanding clearly presenting in the scientific works of Fermat, allowed him still in those distant times to achieve results that for other scientists were either fraught with extreme difficulties or even unattainable. It may be given also the broader definition the notion of number, namely:
A number is a kind of data represented as a function.
This
In a letter from Fermat to Mersenne from 06/15/1641 the following is reported: “
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: (a2+b2)×(c2+d2)=(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 = 142 + 52 = 102 + 112; Now if 2216 = (2213)2 = 107938612; then the required result will be 2217 = (142 + 52)×107938612 = (14×10793861)2 + (5×10793861)2 = 1511140542 + 539693052 = (102 + 112)×107938612=(10×10793861)2 + (11×10793861)2=1079386102 + 1187324712; But you can go also the other way if you submit the initial numbers for example, as follows: 2212 = (142 + 52)×(102 + 112) = (14×10 + 5×11)2 + (14×11 − 5×10)2 = (14×10 − 5×11)2 + (14×11+5×10)2 = 1952 + 1042 = 852 + 2042; 2213 = 2212×221 = (1952 + 1042)×(102 + 112) = (195×10 + 104×11)2 + (195×11 − 104×10)2 = (195×10 − 104×11)2 +(195×11 + 104 × 10)2 = 3 0942 + 11052 = 8062 + 31852; 2214 = (1952 + 1042)×(852 + 2042) = (195×85 + 104×204)2 + (195×204 − 85×104)2 = (195×85 − 104×204)2 + (195×204 + 85×104)2 = 377912 + 309402 = 46412 + 486202; 2217 = 2213×2214 = (30942 + 11052)×(377912 + 309402) = (3094×37791 + 1105×30940)2 + (3094×30940 − 1105×37791)2 = (3094×37791 − 1105×30940)2 + (3094×30940 + 1105×37791)2; 2217 = 1511140542 + 539693052 = 827366542 + 1374874152
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).
For mathematicians and programmers, the notion of function argument is quite common and has long been generally accepted. In particular, f (x, y, z) denotes a function with variable arguments x, y, z. The definition of the essence of a number through the notion of function arguments makes it very simple, understandable and effective since everything what is known about the number, comes from here and all what this definition does not correspond, should be questioned. This is not just the necessary caution, but also an effective way to test the strength of all kinds of structures, which quietly replace the essence of the number with dubious innovations that make science gormlessly and unsuitable for learning.