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

Pic. 17. Karl Friedrich Gauss

Yeah, this was need to be seen how the whole civilized scientific world has driven itself into a dead end! It is obvious that for science, which does not rely on arithmetic, there are no reasonable restrictions and the consequences will be sad – from the dominance of algebra, arithmetic will become so difficult that witlings will call it a science for the elitist mathematicians where they can demonstrate the sharpness of their mind! But the scientists themselves unsuspecting and full of the best intentions, continued to advance science forward to new heights, but so diligently that they either inadvertently or due to a misunderstanding… simply have lost the Fermat's Golden Theorem (FGT)! But this was one of the most impressive discoveries of Pierre Fermat in arithmetic, of which he was very proud.

Pic. 18. Joseph Lagrange

It was so happened that the third in the history royal mathematician Joseph Lagrange together with his predecessor the second royal (and the first imperial!) mathematician Leonard Euler, have proven in 1772 only one special case of FGT for squares and became famous for all the world. This remarkable achievement of science was called the “Lagrange's Theorem about Four Squares”. Probably it is good that Lagrange didn’t live after two years until the moment when in 1815 still very young Augustin Cauchy presented his general proof of the FGT for all polygonal numbers. But then suddenly something terrible happened, the unholy appeared from nowhere and put his "fe" in. And here isn't to you any world fame and besides, you get complete obstruction from colleagues.

Pic. 19. Augustin Cauchy

Well, nothing can be done here, academicians did not like Cauchy and they achieved that this general proof of the FGT was ignored and did not fall into the textbooks as well as no one remembers the Gauss' proofs of 1801 for triangles and for the same squares, nevertheless in the all textbooks as before and very detailed the famous Lagrange's theorem is given. However, after Google published a facsimile of the Cauchy proof of FGT [3], it became clear to everyone why it was not supported by academics (see pt. 3.4.2).

Pic. 20. Marie-Sophie Germain

In the meantime, scientists from around the world inspired by these grand shifts, have so perked up that they wanted overcome the very FLT! They were joined by another famous woman very well known among scientists and mathematicians Marie-Sophie Germain. This talented and ambitious Mademoiselle proposed an elegant way, which was used by at once two giants of mathematical thought Lejeune Dirichlet and Adrien Legendre to prove … only one special case of FLT for the fifth power.

Another such giant Gabriel Lame managed to do the almost impossible and get proof of the highest difficulty … of another particular case of FLT for the seventh power. Thus, the whole elitist quad of the representatives from the high society of scientists was able to prove whole two (!) particular cases of FLT [6, 38].

Pic. 21. Lejeune Dirichlet

Pic. 22. Adrien Legendre

Pic. 23. Gabriel Lame

This result could have been proud since even Euler was also able to prove only two particular cases of FLT for 3rd and 4th powers. In the proof for the 4th power he has applied the descent method following exactly the recommendations of Fermat (see Appendix II). This case is especially important because its proof is valid for all even powers i.e. to obtain a general proof of FLT only odd powers can be considered.

It should be noted that namely Euler has solved (and even significantly expanded!) almost all the most difficult Fermat's tasks and if not for him, then the name Fermat alone could cause real chills to mathematicians. But just not to Sophie Germain who was not at all satisfied the situation with the unproven FLT and she even ventured to suggest that Gauss himself should take up this task! But he simply waved away her replying that the FLT is of little interest to him and such statements, which can neither be proven nor refuted can be found as many as you like.

Of course, Gauss himself would be happy to serve this lady, but if he could do this then it would not need to persuade him. For example, with the help of the “Deductions' arithmetic” developed by Gauss, the prototype of which was the “The Fermat's little theorem”, it was clearly shown how may be to solve the most difficult problems of arithmetic effectively. In particular, only Gauss managed to find a solution to the Fermat's task of calculating two the only squares, the sum of which is a given prime number of types 4n+1 [11, 25].

A characteristic feature of Gauss is his dislike for dubious innovations. For example, he could hardly imagine himself the creator of the geometry of curved spaces. But when he established that such geometry could take place and not contain contradictions, he was very puzzled by this. He was sure that his find could not be of practical use due to the absence of any real facts confirming something like that. However, he quickly found a good way out – he just helped to publish this discovery to his Russian colleague Nikolai Lobachevsky and have done it so skillfully that no one was even surprised when a Russian professor and rector of Kazan University have published a work on non-Euclidean geometry … in Berlin and in German! In the future, Gauss' doubts were confirmed. Followers appeared and flooded science with a whole bunch of similar "discoveries".

Despite the fact that with his proof of the “Basic Theorem of Algebra” Gauss supported Euler in promoting his idea of using “complex numbers”, he did not find any other opportunities for progress in this direction. And what Euler showed, he was also not impressed. Moreover, even modern science at all can nothing offer anything on the use of “complex numbers”. But the sea of all kinds of “scientific” works, studies and textbooks on this theme is clearly inadequate with its true value. Gauss felt that something was amiss with these “numbers” and that it would not end well, therefore in that direction he did not work.

Pic. 24. Ernst Kummer

Thunder struck in 1847 when at a meeting of members of the French Academy of Sciences Gabriel Lame and Augustin Cauchy reported that their FLT proofs was ready for consideration at the competition. However, when in order to identify the winner, it was already possible to open received from them the sealed envelopes, the German mathematician Ernst Kummer having put all scientists on the sinful earth. In his letter it was reported that the FLT proof on the basis of “complex numbers” is impossible due to the ambiguity of their decomposition into prime factors. 17 17 According to the Basic theorem of arithmetic the decomposition of any natural number into prime factors is always unambiguous, for example, 12=2×2×3 i.e. with other prime factors this number like any other, is impossible to imagine. But for “complex numbers” in the general case this unambiguity is lost for example, 12=(1+√–11)×(1+√–11)=(2+√–8)×(2+√–8) In fact, this means the collapse of science in its very foundations. However, the generally accepted criteria (in the form of axioms) what can be attributed to numbers and what is not, as there was not so still is not.

Here you have got what you want! These very “complex numbers” are not any numbers!!! And one could notice finally, after arithmetic was knocked from under science, it hangs in the air having no solid foundation. And the mistakes of the greats in their consequences are also extreme and they begin to break down a science so much that, instead of a holistic system of knowledge, it creates a bunch of unrelated fragments.

If it so happened, then else in 1847, these very “Complex numbers” had to be solemnly buried with all the honors. But with this matter somehow did not work out at all and the restless souls of the long-dead theories turn out to be so tenacious that they cannot be expelled from textbooks and professorial lectures by any means. They will wander through different books and reference books whose authors will be completely unaware of how much their works depreciates from this useless ballast.

In the mentioned book of Singh is well shown as the ambiguity of decomposing compound integers into prime factors makes it impossible to construct logical conclusions in proofs and it also was said that the unambiguity theorem for such a decomposing for natural numbers was given in “Euclidean Elements”. The specific book and location of the theorem is not specified; therefore, it is rather difficult to find the necessary text, but it really turned out to be so. 18 18 The theorem and its proof are given in “The Euclid's Elements” Book IX, Proposition 14. Without this theorem, the solution of the prevailing set of arithmetic problems becomes either incomplete or impossible at all.

“Euclidean Elements" is a very old book with archaic terminology, in which this extremely important for science theorem was somehow lost and it was simply forgotten about it. The first to discover the loss was Gauss. He formulated it again and gave proof, which contained a surprisingly simple and even childish error, where as an argument used exactly what needs to be proven (see pt. 3.3.1).

But this is not an ordinary theorem, all science holds on it! And what about Euclid? Oh my God! In fact, his proof is the same as that of Gauss i.e. wrong!!! Tell it to someone, so they will not believe! Three giants of science are stumbled on the same place!

Pic. 25. Euclid

Then it turns out that this whole science is fake and now, thanks to Singh’s book and despite all the good intentions of its author, this terrifying FLT, which now even in theory has become completely unprovable, was so furious that like a true monster, in one fell swoop have devalued all the age-old works of scientists! And yet they live in not fabulous, but in the real kingdom of crooked mirrors, what about they themselves don’t know anything.