Sequences of consecutive squares on elliptic curves and elliptic curves with high rank /
Mohammed Gamal Kamel Asraan
Sequences of consecutive squares on elliptic curves and elliptic curves with high rank / متتابعات من التربيعات المتتالية على المنحنيات الناقصة و المنحنيات الناقصة ذات الرتب العالية Mohammed Gamal Kamel Asraan ; Supervised Nabil L. Youssef , Mohammad M. Sadek - Cairo : Mohammed Gamal Kamel Asraan , 2017 - 82 P. ; 25cm
Thesis (M.Sc.) - Cairo University - Faculty of Science - Department of Mathematics
In this thesis, we investigate a certain kind of progressions on elliptic curves, namely, sequences of consecutive squares. Let C be an elliptic curve dened over Q by the equation y² = f(x), where f(x) Q[x] is a polynomial of degree 3 or 4. A sequence of rational points (x₁, y₂) C(Q), i =1, 2, . . . , is said to form a sequence of consecutive squares on C if the sequence of x-coordinates, xi, i =1, 2, . . ., consists of consecutive rational squares. We produce an innite family of elliptic curves dened by the equation C : y² =x³+A₂+B with a 5-term sequence of consecutive squares and we prove that the rational points in this sequence are independent. Moreover, we display an innite family of elliptic curves dened by the equation E : y² =Ax⁴+Bx²+C possessing a 6-term sequence of consecutive squares and prove that these six terms are independent in E(Q). On the other hand, we investigate the rank of elliptic curves with torsion group isomorphic to one of the groups Z₇, Z₉, Z₁₀, Z₁₂ or Z₂Z₈. For elliptic curves with torsion group isomorphic to Z₇, we produce an innite family of such elliptic curves with rank 2. Furthermore, for each group of Z₉, Z₁₀, Z₁₂ and Z₂Z₈, we construct an innite family of elliptic curves of rank 1 and torsion group isomorphic to one of the aforementioned groups. All the constructed elliptic curves are parameterized by rational points of an elliptic curve with positive rank
Consecutive squares Elliptic curves High rank
Sequences of consecutive squares on elliptic curves and elliptic curves with high rank / متتابعات من التربيعات المتتالية على المنحنيات الناقصة و المنحنيات الناقصة ذات الرتب العالية Mohammed Gamal Kamel Asraan ; Supervised Nabil L. Youssef , Mohammad M. Sadek - Cairo : Mohammed Gamal Kamel Asraan , 2017 - 82 P. ; 25cm
Thesis (M.Sc.) - Cairo University - Faculty of Science - Department of Mathematics
In this thesis, we investigate a certain kind of progressions on elliptic curves, namely, sequences of consecutive squares. Let C be an elliptic curve dened over Q by the equation y² = f(x), where f(x) Q[x] is a polynomial of degree 3 or 4. A sequence of rational points (x₁, y₂) C(Q), i =1, 2, . . . , is said to form a sequence of consecutive squares on C if the sequence of x-coordinates, xi, i =1, 2, . . ., consists of consecutive rational squares. We produce an innite family of elliptic curves dened by the equation C : y² =x³+A₂+B with a 5-term sequence of consecutive squares and we prove that the rational points in this sequence are independent. Moreover, we display an innite family of elliptic curves dened by the equation E : y² =Ax⁴+Bx²+C possessing a 6-term sequence of consecutive squares and prove that these six terms are independent in E(Q). On the other hand, we investigate the rank of elliptic curves with torsion group isomorphic to one of the groups Z₇, Z₉, Z₁₀, Z₁₂ or Z₂Z₈. For elliptic curves with torsion group isomorphic to Z₇, we produce an innite family of such elliptic curves with rank 2. Furthermore, for each group of Z₉, Z₁₀, Z₁₂ and Z₂Z₈, we construct an innite family of elliptic curves of rank 1 and torsion group isomorphic to one of the aforementioned groups. All the constructed elliptic curves are parameterized by rational points of an elliptic curve with positive rank
Consecutive squares Elliptic curves High rank