This document contains exercises on number theory, focusing on coprime numbers and irrational roots, tailored for students specializing in mathematics at the Terminal S level.
Extract
[...] In all other cases, is irrational. Exercice 40 One obtains: One checks that two consecutive terms of this sequence are always coprime. One has: Thus, according to Bezout's theorem, and are first among them. The 4 first terms are: . Two consecutive terms of this sequence are indeed first among them. On It is deduced from this last equality that the GCD of and divise 2. But since they are always odd numbers cannot be a divisor. Thus the GCD is These two numbers are coprime . [...]
[...] So, by Bezout's theorem, there exist two relative numbers and such as: . On a then and so . We obtain: On deduce from Bezout's theorem that and are first among them. If we apply this last result to the couple here are prime between them, we can deduce that and are first among them. On a Or divise so divise also On Thus divise . Or d'après l'énoncé, and are first among them. So according to question and are first among them also. [...]
