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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 ca
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
“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.
The fiasco being by academicians Cauchy and Lame did not result in the rejection use of the surrogates of numbers in science especially after Kummer who had crushed their works, found a way to prove FLT (with a little modernization) for any particular case. Before the final victory over the FLT only a last step remained – to obtain a single common proof. Since then 170 years have passed, but nothing was changed. Supported in due time by the Euler's genius "complex numbers" are still presented today as a kind of extension the notion of number. This looks very impressive and solid, but still requires a clear definition of the very notion of number, however just with this deal are very bad.
Students intuitively feeling that they are being tortured in vain by nonsenses about some non-existent numbers, suddenly have a question: “What is a number?” They never come to mind that not a single professor could not answer this question even if he has reread everything that is in mathematics. One of them even could not bear the mocking hints and had published a whole book called “What is a number?” [13, 29]. In it, he has written so many whatnots that students have very well understood – such a question it’s better not to ask.
Pic. 26. Francis Viète
Meanwhile, scientists continued to move science forward, not bothering with such trifles as the essence of the notion of number. So, they created a whole bunch of new algebras taking advantage of the fact that there were no obstacles along the way. But they were not a continuation of what was a real one, the founder of which was the first royal mathematician François Viète served as an advisor to the court of the French king Henry III. But if these new algebras are special, then their terminology and bases are also special.
So, little by little in the science began to form a particular bird language understandable only to the authors of these most i
Georg Cantor has developed his theory of sets, which other mathematicians such as, for example, Henri Poincaré, called all sorts of bad words and did not want to admit at all. But suddenly unexpected for everyone the respectable "Royal Society of London" (the English Academy of Sciences) in 1904 decided to award Cantor with its medal. So, it turns out that here is the point, where the fates of science are decided!20
Pic. 27. Georg Cantor
And everything would be fine, but suddenly another trouble struck again. Out of nowhere in this very theory of sets insurmountable contradictions began to appear, which are also described in great detail in Singh’s book. In the scientific community everyone immediately was alarmed and began to think about how to solve this problem. But it has rested as on the wall and in no way did not want to be solved. Everyone was somehow depressed, but then they yet cheered up again.
It was so happened because now David Hilbert himself got down to it, the great mathematician that first solved the very difficult Waring problem, which has a direct relationship to the FLT. 21 It is also curious that Hilbert repeated Euler's experiment apparently inspired by the FLT problem. It seems that at some point Euler began to have doubts that the FLT is generally provable and he assumed the equation a4+b4+c4=d4 also like Fermat’s equation an+bn=cn for n>2 in integers is unsolvable, but in the end it turned out that he was wrong.22
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.
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.
19
Soviet mathematician Lev Pontryagin showed these “numbers” do not have the basic property of commutativity i.e. for them ab ≠ ba [34]. Therefore, one and the same such “number” should be represented only in the factorized form, otherwise it will have different value at the same time. When in justification of such creations scientists say that mathematicians have lack some numbers, in reality this may mean they obviously have lacked a mind.
20
If some very respected public institution thus encourages the development of science then what one can object? However, such an emerging unknown from where the generosity and disinterestedness from the side of the benefactors who didn’t clear come from, looks somehow strange if not to say knowingly biased. Indeed, it has long been well known where these “good intentions” come from and whither they lead and the result of these acts is also obvious. The more institutions there are for encouraging scientists, the more real science is in ruins. What is costed only one Nobel Prize for "discovery" of, you just think … accelerated scattering of galaxies!!!
21
Waring's problem is the statement that any positive integer N can be represented as a sum of the same powers xin, i.e. in the form N = x1n + x2n + … + xkn. It was in very complex way first proven by Hilbert in 1909, and in 1920 the mathematicians Hardy and Littlewood simplified the proof, but their methods were not yet elementary. And only in 1942 the Soviet mathematician Yu. V. Li
22
A counterexample refuting Euler’s hypothesis is 958004 + 2175194 + 4145604 = 4224814. Another example 26824404+153656394+187967604=206156734. For the fifth power everything is much simpler. 275+845+1105+1335=1445. 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.