Pravokutni trokuti s racionalnim stranicama zadane površine

Jedan od najstarijih poznatih matematičkih problema odnosi se na određivanje pravokutnih trokuta čije su duljine stranica racionalni brojevi, a površina je jednaka zadanom prirodnom broju \(\mathcal A\). U suvremenoj terminologiji takav broj \(\mathcal A\) naziva se kongruentnim brojem. U ovom radu prikazana je geometrijska metoda kojom se za pojedini kongruentni broj \(\mathcal A\), počevši od jednog pripadnog pravokutnog trokuta s racionalnim duljinama stranica, može iterativnim postupkom konstruirati niz trokuta s racionalnim stranicama i površinom oblika \(\mathcal Ar^2\), pri čemu je \(r\) racionalan broj. Opis skupa \(T_{\mathcal A}\) svih pravokutnih trokuta površine \(\mathcal A\), s racionalnim duljinama stranica, blisko je povezan s grupom \(E_{\mathcal A}(\mathbb{Q})\) racionalnih točaka eliptičke krivulje \(E: y^2 = x^3 - \mathcal A^2x\).Prikladnom parametrizacijom skupa \(T_{\mathcal A}\) i primjenom geometrijske konstrukcije dobivaju se nove racionalne točke na krivulji. Na skupu \(T_{\mathcal A}\) uvodi se algebarska struktura u kojoj se taj skup pokazuje konačno generiranim. U pojedinim primjerima na temelju toga moguće je potpuno opisati skup \(T_{\mathcal A}\), no problem efektivnog algoritma za pronalaženja skupa generatora ostaje otvoren.One of the oldest known problems in mathematical research is to determine all right-angled triangles with rational sidelengths such that their area equals a given positive integer \(\mathcal A\). In modern terminology such a number \(\mathcal A\) is called a congruent number. In this thesis, a geometric method of producing new rational-sided right triangles with area \(\mathcal A\), starting from one given triangle, is exhibited. Iterating this construction, an infinite sequence of nonsimilar triangles with the same property may be obtained. A description of the set \(T_{\mathcal A}\) of all right-angled triangles with rational sidelengths and area \(\mathcal A\) is closely related to the group \(E_{\mathcal A}(\mathbb{Q})\) of rational points of the elliptic curve \(E: y^2 = x^3 - \mathcal A^2x\). By a suitable parametrization of \(T_{\mathcal A}\) and application of the geometric construction, new rational points on \(E\) are obtained. An algebraic structure is introduced on \(T_{\mathcal A}\), in which this set is shown to be finitely generated. Using that fact, in some specific examples it is possible to give a complete description of \(T_{\mathcal A}\), but the problem of an effective algorithm for finding the set of generators remains open