Then we obtain:
AiBiEi=(2m)n (10)
where Ai = c−b=a−2m; Bi=c−a=b−2m; Ei – polynomial of power n−2.
Equation (10) is a ghost that can be seen clearly only on the assumption that the number {an+bn−cn} is reduced when (1) is substituted into (8). But if it is touched at least once, it immediately crumbles to dust. For example, if Ai×Bi×Ei=2m2×2n-1mn-2 then as one of the options could be such a system:
AiBi=2m2
Ei=2n-1mn-2
In this case, as we have already established above, it follows from AiBi=2m2 that for any natural number m the solutions of equation (1) must be the Pythagoras’ numbers. However, for n>2 these numbers are clearly not suitable and there is no way to check any other case because in a given case (as with any other variant with the absence of solutions) another substitution will be definitely unlawful and the ghost equation (10), from which only solutions can be obtained, disappears.61 Since the precedent with an unsuccessful attempt to obtain solutions has already been created, there can be no doubt that also all other attempts to obtain solutions from (10) will be unsuccessful because at least in one case the condition {an+bn−cn}=0 is not fulfilled i.e. the equation (10) has been obtained by substituting a non-existing (1) in the key formula (2), and the Fermat’s Last Theorem is proven.62
So, now we have a restored author's proof of the Fermat's most famous theorem. Here are interesting ideas, but at the same time there is nothing that could not be accessible to science for more than three hundred years. Also, from the point of view a difficulty in understanding its essence, it matches at least to the 8th grade of secondary school. Undoubtedly, the FLT is a very important component of number theory. However, there is no apparent reason that this task has become an unsolvable problem for centuries, even though millions of professional scientists and amateurs have taken part in the search for its solution. It remains now only to lament, that's how he is, this unholy!
After everything was completed so well with the restoration of the FLT proof, many will be disappointed because now the fairy tale is over, the theme is closed and nothing interesting is left here. But this was the case before, when in arithmetic there were only rebuses, but we know that this is not so, therefore for us the fairy tale not only has not ended, but even did not begin! The fact is that we have so far revealed the secret of only two of the Fermat’s six recordings, which we have restored at the beginning of our study. To make this possible, we made an action-packed historical travel, in which the LTF was an extra-class guide. This travel encouraged us to take advantage of our opportunities and look into these forbidden Fermat’s “heretical writings” to finally make a true science in image of the most fundamental discipline of arithmetic available to our intelligent civilization and allowing it to develop and flourish on this heavy-duty foundation like never before.
We can honestly confess that so far not everything that keep in Fermat's cache is accessible and understandable to us. Moreover, we cannot even determine where this place is. But also, to declare that everything that we tell here, is only ours, would be clearly unfair and dishonest because nobody would have believed us then. On the other hand, if everything was so simple, then it would be completely to no one interesting. The worst thing that could be done, is to reveal the entire contents of Fermat’s cache so that everyone will forget about it immediately after reading.
We will act otherwise. If something will be revealed, only to give an opportunity to learn about the even more innermost mysteries of science, which will not only make everyone smarter, but will indicate the best ways to solve vital problems. Using the example of solving the FLT problem, it will be quite easily to make sure this since with a such solution science receives so a reliable point of support that it can do whatever it wants with the integer power numbers. In particular, it may be easily calculated as much as you like of integer power numbers, which in sum or difference will give again an integer power number. The fact that only a computer can shovel such a work, is very ashamed for current science because this task is too simple even for children.
The most quick-witted of them will clearly prefer that adults ask them to explain something more difficult for example, FLT proof, which in their time was completely inaccessible to them. Children of course, will not fail to get naughty and will be important like high-class nobles when answer to stupid questions of adults and indicating to them that it would be nice for someone to learn something else. But it will be only little flowers. But after that, the amazement of adults will become simply indescribable when they find out that the children are addicted to peeping and copying everything that interests them directly from Fermat's cache! Indeed, at their age they still do not realize their capabilities and it seems to them that this is at all not a difficult task.
However, if they had not read interesting books about science, then such an idea would never have occurred to them. But when they find out that someone is doing this, they will find that they can do it just as well if even not better! Do you not believe? Well, everyone who wants to be convinced of this, will have this opportunity now. But one more small detail remains. Fermat in his "heretical writings" although he pointed out that he had to provide proofs of three simple theorems for children, which he specially prepared for them, but so far he did not have time for this nevertheless firmly promised that as soon as he has a time, then he will certainly and sure do it.
But apparently, he did not have enough time and so he did not manage to add the necessary recordings. Or perhaps he changed his mind because didn’t want to deprive children of joy on their own to learn to solve just such problems that adults can't afford. If the children can't cope, then who them will reproach for it. But if they manage it, then adults will not go anywhere and will bring to children many, many gifts!
4.3. Theorems About Magic Numbers
The above presented proof of FLT not only corresponds to Fermat's assessment as" truly amazing", but is also constructive since it allows us to calculate both the Pythagorean numbers and other special numbers in a new way what demonstrate the following theorems.
Theorem 1. For any natural number n, it can be calculated as many
triples as you like from different natural numbers a, b, c such that
n = a2 + b2 – c2. For example :
n=7=62+142–152=282+1282–1312=5682+51882–52192=
=1783282+53001459282–53001459312 etc.
n=34=112+132–162=3232+30592–30762=
=2475972+20434758052–20434758202 etc.
The meaning of this theorem is that if there is an infinite number of Pythagoras triples forming the number zero in the form a2+b2−c2=0 then nothing prevents creating any other integer in the same way. It follows from the text of the theorem that numbers with such properties can be “calculated”, therefore it is very useful for educating children in school.
