Читать книгу The Wonders of Arithmetic from Pierre Simon de Fermat - Youri Veniaminovich Kraskov - Страница 18

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 = a² + b² – c². For example :

n=7=6²+14²–15²=28²+128²–131²=568²+5188²–5219²=

=178328²+5300145928²–5300145931² etc.

n=34=11²+13²–16²=323²+3059²–3076²=

=247597²+2043475805²–2043475820² etc.

The meaning of this theorem is that if there is an infinite number of Pythagoras triples forming the number zero in the form a²+b²−c²=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.