A dynamical analogue of a question of fermat

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Sadek, Mohammad and Yesin Elsheikh, Emine Tuğba (2024) A dynamical analogue of a question of fermat. Mathematica Scandinavica, 130 (1). pp. 5-20. ISSN 0025-5521

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Abstract

Given a quadratic polynomial with rational coefficients, we investigate the existence of consecutive squares in the orbit of a rational point under the iteration of the polynomial. We display three different constructions of 1-parameter quadratic polynomials with orbits containing three consecutive squares. İn addition, we show that there exists at least one polynomial of the form x2 + c with a rational point whose orbit under this map contains four consecutive squares. This can be viewed as a dynamical analogue of a question of Fermat on rational squares in arithmetic progression. Finally, assuming a standard conjecture on exact periods of periodic points of quadratic polynomials over the rational field, we give necessary and sufficient conditions under which the orbit of a periodic point contains only rational squares.
Item Type: Article
Divisions: Faculty of Engineering and Natural Sciences
Depositing User: Mohammad Sadek
Date Deposited: 09 Jun 2024 22:07
Last Modified: 09 Jun 2024 22:07
URI: https://research.sabanciuniv.edu/id/eprint/49243

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