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 ofnumber, as well as a new versionsaxioms 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>2 by 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
