Irrational numbers - Colegiul Național din Iași (Iași - Roumanie)

Titre du sujet
Irrational numbers
Établissement
Colegiul Național din Iași (Iași - Roumanie)
Année
2020-2021
Résumé
Let d > 1 be an integer. Determine all numbers of the form a+b√d with a et b integers such that their inverses are of the same form (a+b√d with a et b integers). For instance, for d = 2, 1+ √2 has this property, since: (-1+√2)(1+√2)=1.

One can assume d is squarefree (d ≠ 0 and 1, and d is not divisible by the square of any prime; d can be 2, 3, 5, 6,..., but not 8 or 18).
Mots clés