Glasnik Matematicki, Vol. 52, No. 1 (2017), 4552.
ON SEQUENCES OF CONSECUTIVE SQUARES ON ELLIPTIC CURVES
Mohamed Kamel and Mohammad Sadek
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
email: mohgamal@sci.cu.edu.eg
Mathematics and Actuarial Science Department, American University in Cairo, AUC Avenue, New Cairo, Egypt
email: mmsadek@aucegypt.edu
Abstract.
Let C be an elliptic curve defined over Q by the equation y^{2}=x^{3}+Ax+B where A,BQ. A sequence of rational points (x_{i},y_{i}) C(Q), i=1,2,…, is said to form a sequence of consecutive squares on C if the sequence of xcoordinates, x_{i},i=1,2,…, consists of consecutive squares. We produce an infinite family of elliptic curves C with a 5term sequence of consecutive squares. Furthermore, this sequence consists of five independent rational points in C(Q). In particular, the rank r of C(Q) satisfies r≥ 5.
2010 Mathematics Subject Classification.
14G05, 11B83.
Key words and phrases. Elliptic curves, rational points, sequences of consecutive squares.
Full text (PDF) (access from subscribing institutions only)
DOI: 10.3336/gm.52.1.04
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