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

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 = a 2+ b 2 – c 2. For example :

n=7=6 2+14 2–15 2=28 2+128 2–131 2=568 2+5188 2–5219 2=

=178328 2+5300145928 2–5300145931 2etc.

n=34=11 2+13 2–16 2=323 2+3059 2–3076 2=

=247597 2+2043475805 2–2043475820 2etc.

The meaning of this theorem is that if there is an infinite number of Pythagoras triples forming the number zero in the form a 2+b 2−c 2=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.

In this case, we will not act rashly and will not give here or anywhere else a proof of this theorem, but not at all because we want to keep it a secret. Moreover, we will recommend that for school books or other books (if of course, it will appear there) do not disclose the proof because otherwise its educational value will be lost and children who could show their abilities here will lose such an opportunity. On the other hand, if the above FLT proof would remain unknown, then Theorem 1 would be very difficult, but since now this is not so, even not very capable students will quickly figure out how to prove it and as soon as they do, they will easily fulfill the given above calculations.

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Naturalized geometric elements form either straight line segments of a certain length or geometric figures composed of them. To make of them figures with curvilinear contours (cone, ellipsoid, paraboloid, hyperboloid) is problematic, therefore it is necessary to switch to the representation of geometric figures by equations. To do this, they need to be placed in the coordinate system. Then the need for naturalized elements disappears and they are completely replaced by numbers for example, the equation of a straight line on the plane looks as y=ax+b, and the circle x 2+y 2=r 2, where x, y are variables, a, b are constants offset and slope straight line, r is the radius of the circle. Descartes and independently of him Fermat had developed the fundamentals of such (analytical) geometry, but Fermat went further proposing even more advanced methods for analyzing curves that formed the basis of the Leibniz – Newton differential and integral calculus.

Under conditions when the general state of science is not controlled in any way, naturally, the process of its littering and decomposition is going on. The quality of education is also uncontrollable since both parties are interested in this, the students who pay for it and the teachers who earn on it. All this comes out when the situation in society becomes conflict due to poor management of public institutions and it can only be “rectified” by wars and the destruction of the foundations of an intelligence civilization.

The name itself “the Basic theorem of arithmetic”, which not without reason, is also called the Fundamental theorem, would seem a must to attract special attention to it. However, this can be so only in real science, but in that, which we have, the situation is like in the Andersen tale when out of a large crowd of people surrounding the king, there is only one and that is a child who noticed that the king is naked!

On a preserved tombstone from the Fermat’s burial is written: “qui literarum politiforum plerumque linguarum” – skilled expert in many languages (see Pic. 93-94 in Appendix VI).

It is believed that Fermat left only one proof [36], but this is not entirely true since in reality it is just a verbal description of the descent method for a specific problem (see Appendix II).

It was a truly grandiose mystification, organized by Princeton University in 1995 after publishing in its own commercial edition "Annals of Mathematics" the “proof” of FLT by A. Wiles and the most powerful campaign in the media. It would seem that such a sensational scientific achievement should have been released in large numbers all over the world. But no! Understanding of this text is available only to specialists with appropriate training. Wow, now even that, which cannot be understood, may be considered as proof! However, for fairness it should be recognized that even such an overtly cynical mockery of science, presented as the greatest "scientific achievement" of the luminaries of Princeton University, cannot be even near to the brilliant swindle of their countrymen from the National Space Administration NASA, which resulted that the entire civilized world for half a century haven’t any doubt that the American astronauts actually traveled to the moon!

The “proof”, which A. Wiles prepared for seven years of hard work and published on whole 130 (!!!) journal pages, exceeded all reasonable limits of scientific creativity and of course, him was awaiting inevitable bitter disappointment because such an impressive amount of casuistry understandable only to its author, neither in form nor in content is in any way suitable to present this as proof. But here the real wonder happened. Suddenly, the almighty unholy himself was appeared! Immediately there were influential people who picked up the "brilliant ideas" and launched a stormy PR campaign. And here is your world fame, please, many titles and awards! The doors to the most prestigious institutions are open! But such a wonder even for the enemy not to be wish because sooner or later the swindle will open anyway.

If this book was published during the life of Fermat, then he would simply be torn to pieces because in his 48 remarks he did not give a proof of any one of his theorems. But in 1670 i.e. 5 years after his death, there was no one to punish with and venerable mathematicians themselves had to look for solutions to the problems proposed by him. But with this they obviously had not managed and of course, many of them could not forgive Fermat of such insolence. They were also not forgotten that during his lifetime he twice arranged the challenges to English mathematicians, which they evidently could not cope with, despite his generous recognition of them as worthy rivals in the letters they received from Fermat. Only 68 years after the first publication of Diophantus' "Arithmetic" with Fermat's remarks, did the situation at last get off the ground when the greatest science genius Leonard Euler had proven a special case of FLT for n=4, using the descent method in exact accordance with Fermat's recommendations (see Appendix II). Later thanks to Euler, there was received solutions also of the other tasks, but the FLT had so not obeyed to anyone.