However, if they had not read interesting books about science, then such an idea would never have occurred to them. But when they find out that someone is doing this, they will find that they can do it just as well if even not better! Do you not believe? Well, everyone who wants to be convinced of this, will have this opportunity now. But one more small detail remains. Fermat in his "heretical writings" although he pointed out that he had to provide proofs of three simple theorems for children, which he specially prepared for them, but so far he did not have time for this nevertheless firmly promised that as soon as he has a time, then he will certainly and sure do it.

But apparently, he did not have enough time and so he did not manage to add the necessary recordings. Or perhaps he changed his mind because didn’t want to deprive children of joy on their own to learn to solve just such problems that adults can't afford. If the children can't cope, then who them will reproach for it. But if they manage it, then adults will not go anywhere and will bring to children many, many gifts!

The above presented proof of FLT not only corresponds to Fermat's assessment as" truly amazing", but is also constructive since it allows us to calculate both the Pythagorean numbers and other special numbers in a new way what demonstrate the following theorems.

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^>2 etc.

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

=247597^>2+2043475805^>2–2043475820^>2 etc.

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.