Glasnik Matematicki, Vol. 52, No. 1 (2017), 45-52.

ON SEQUENCES OF CONSECUTIVE SQUARES ON ELLIPTIC CURVES

Mohamed Kamel and Mohammad Sadek

Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
e-mail: mohgamal@sci.cu.edu.eg

Mathematics and Actuarial Science Department, American University in Cairo, AUC Avenue, New Cairo, Egypt
e-mail: mmsadek@aucegypt.edu


Abstract.   Let C be an elliptic curve defined over Q by the equation y2=x3+Ax+B where A,BQ. A sequence of rational points (xi,yi) 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 squares. We produce an infinite family of elliptic curves C with a 5-term 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.


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DOI: 10.3336/gm.52.1.04


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