The Wonders of Arithmetic from Pierre Simon de Fermat - страница 52

Then we obtain:

A_>iB_>iE_>i=(2m)^>n (10)

where A_>i = c−b=a−2m; B_>i=c−a=b−2m; E_>i – polynomial of power n−2.

Equation (10) is a ghost that can be seen clearly only on the assumption that the number {a^>n+b^>n−c^>n} is reduced when (1) is substituted into (8). But if it is touched at least once, it immediately crumbles to dust. For example, if A_>i×B_>i×E_>i=2m^>2×2^>n-1m^>n-2 then as one of the options could be such a system:

A_>iB_>i=2m^>2

E_>i=2^>n-1m^>n-2

In this case, as we have already established above, it follows from A_>iB_>i=2m^>2 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 {a^>n+b^>n−c^>n}=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.