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

Within the framework of the designated topic:

1. There are restored some little-known facts from the Pierre Fermat's biography.

2. For the first time, there are given two definitions (mathematical and general) the concept of number , as well as a new versions axioms and basic properties of numbers following from them.

3. There is shown fallacy of Euclid, Gauss, and Zermelo's proofs of the Basic Theorem of Arithmetic (BTA), without which the foundation of the whole science collapses.

4. There are given comments on Zermelo's proof of the Basic Theorem of Arithmetic.

5. There is restored the Fermat’s BTA proof by the descent method .

6. There is restored a simplest method of proving the FLT for the 4th power.

7. There is shown the fallacy of the G. Frey’s idea lied in the basis of the A. Wiles’ FLT proof 1995.

8. There is restored the Fermat’s proof of his Last Theorem on the basis of a new way to solving the Pythagoras’ equation x 2+y 2=z 2by using the formula, discovered by him and called Fermat Binomial .

9. As a consequence of the FLT proof there are formulated Theorems on Magic Numbers , the validity of which is confirmed by examples of calculations.

10. There is proposed the formulation of the Beal Theorem revealing the essence of the Beal Conjecture for equation A x+B y=C z. There are given examples of calculations according to this theorem.

11. There is restored a way for proving the Fermat's Golden Theorem .

12. There is restored the proof’s method of the Fermat's grandiose discovery about primes of the form 4n+1.

13. There is restored a way of solving the Archimedes-Fermat equation Ax 2+1=y 2.

14. There is restored a proof of the Fermat's theorem on the unique solution of the equation y 3=x 2+2

15. There are shown examples to application of methods for solving problems proposed by Fermat.

16. There is shown a role of arithmetic as the basis of foundations of the whole science.

17. There are shown examples of non-existent sciences such as history, informatics and economics.

18. There are given definitions the essence of the basic concepts’ informatics and economics .

19. For the first time, there is given a general definition the essence of the concept an information .

20. There are proposed some fundamentals of informatics as a science .

21. There is proposed a method for ordering knowledge using the Basic law of systems .

22. There is proposed the idea of an economic breakthrough based on the new generation of IT .

23. There is proposed the new essential understanding of money and their functions .

24. There is proposed an International Payment System (IPS) of a fundamentally new type using national currencies in international settlements.

25. There is given essential understanding the source of profit on invested capital .

26. Historical episodes are presented in a stylized literary form.

27. There is proposed the restoration of the tombstone of P. Fermat with an English translation.

28. There are proposed 15 riddles in Fermat-style i.e. without their complete solution.

29. There is compiled a full list of scientific problems presented in this book in 100 points.

30. The original Russian text of the book is translated by its author into English.

The book is intended for a wide range of readers.

Feedback and comments may be sent to c_city2000@mail.ru

Imagine that you decided to write a book and such an interesting one so that, wow, it is breathtaking! But how to begin? It's very simple, you open this book and immediately see the famous picture with the portrait of our main hero the legendary Pierre de Fermat from the complete collection of all his written works, that Paul Tannery and Charles Henry released at the beginning of the 20th century. So here this picture is now decorated with a wonderful equation called Fermat’s binomial formula, about which the current science still has no idea, although this formula was published back in 2008.

Pic. 1. Portrait of the Senator Pierre de Fermat

And the next picture is a delightful sculptural composition. Our hero is so good-looking and next to him is his muse, from her he drew his inspiration and created such wonders, from which the entire civilized world since the XVII century and until now just goes crazy.

Pic. 2. Pierre Fermat and His Muse in the Capitol of Toulouse

After this picture follows the image of a page from Diophantus' “Arithmetic” with the text of the Fermat’s Last Theorem, published in 1670, and from here immediately the main theme appears.

Pic. 3. Diophantus’s “Arithmetic” Page of the Greek-Latin Edition 1670 with the Task VIII and a Remark to it, Becoming Later the Fermat’s Last Theorem

But this is about science and can it really be something interesting?

In the one, that we are taught, it is hardly, but in the true science there can be really amazing wonders! Because, unlike conventional belles-lettres, science is not just a literary embodiment of the author’s idea, everything here is much more complicated, since he should reflect not invented by him, but the truest reality, which can always be checked and if something is wrong, then the whole work will go down the drain. Such are the cruel laws of this genre. However, really, if you just look at the content of the book … there are some problems, tasks, refutations … outwardly it looks somehow not very intriguing. But the impression will change when we after the two wonderful pictures at once take the bull by the horns – we give another picture by our hero with his really written marks in the margins of one old book …

Pic. 4. Fermat’s Recording in Margins of Apollonius the Perganos '

Book, named Conic Section

and right behind it suddenly … bubuh!!!

Pic. 5. The Page of the Diophantus’ “Arithmetic” Edition 1621 and the Restored Text of the Fermat Last Theorem in the Margins

Oh, what is it??? … This is just not possible!!! Even eyes cannot believe. A page from Diophantus’ “Arithmetic” only an earlier edition 1621 from Fermat’s personal copy of the book with the most famous theorem on the margins, from which the entire scientific world is still in a fever! Everyone knows this book has been lost! But where does it come from?

From there! The book was so well hidden that so far no one knew about it!

Yeah … and what's next?

The next is that Fermat’s binomial formula is from this book of ours, where it will be derived, and it is easily to make sure that formula is true. Take any two numbers x, y and the third z their sum (or difference), substitute in the formula and everything will be OK! Isn't that curious?

Well, but what is the avail of this formula and what is its significance?

But without it the Fermat Last Theorem will remain unproven!

Yeah… would Fermat really hide such a treasure from everyone?

Well, that's a good question. Anyway, the plot is already twisted in the right direction and quite famously … although this is not even the beginning and only a light aperitif, but it acting already is oh so good!

Is it true that a lot of money was promised for this theorem?

Hmm, promised – yes, and even paid … but not at all for the FLT!

How is this so, and for what?

For the error, that was revealed in the proof, but science did not want notice it for 25 years.

So, it turns out all this work with our book will go past the cash box?

Well, that's another good question. However, it has already long been known that for science money is not paid. Well, if … as if for … doing science … or taking care of it … well, for the good of the state, then yes, billions are spinning here, but the very science has never existed and does not exist now.

So perhaps we need to do something incredible here in this book so that we can get something out of these billions?

But no, then no one will read this book, besides all the places of the distribution of money have long been occupied. It is necessary to do this book with real science, but for this we need to use special approaches and principles. Particularly, the main principle should be that all the topics touched upon here contain secrets, and such that they are hidden not otherwise … as it is said … ah yes, behind seven seals!

Wow, it’s light to say…

Moreover, about this should be announced in advance (we are now doing this) that even in relatively calm places of narration no one should have any doubt that in every topic something must necessarily be very much surprise or even shake! Well, not literally, of course. Say, it is given in some place a calm explanation of some entity or their components and suddenly, unexpectedly … bubuh!!! And it turns out that here the amazing secret is hiding, which was lurking under the guise of what has been known for a long time, but did not attract any attention. Another principle is that of all the secrets we have revealed (and there are more than a hundred of them here!!!) we must leave some share (say, one and half dozen) as undisclosed. Otherwise the children will be offended that them no one nothing left for their own discoveries …

!!!… What?… The children???

Oh yes, we narrowly forgot to say that this book is largely childish. Most adults would be unable to retrain for a such a thorough science. It is clear, they will very offended hearing this, but we of course will not trumpet about it over the entire world, and even will observe decency i.e. from all the tasks of basic science that we will discuss, only the lightest will be singled out specifically for children. They’ll be happy about this because among these tasks there will be … the very FLT!!! Yes, it is in this book of ours that for the first time in the entire history of the FLT, this problem, and not just a problem, but one of the fundamental basics of science (!!!) will be solved in expanded form together with some other problems related to it. In this sense, the FLT will be our guiding star and its first-discoverer Pierre Fermat will be our mentor from whom we will directly (!) receive all the information we need.