@article{Sadek_Yesin_2024, title={A dynamical analogue of a question of Fermat}, volume={130}, url={https://www.mscand.dk/article/view/142342}, DOI={10.7146/math.scand.a-142342}, abstractNote={<p>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. In addition, we show that there exists at least one polynomial of the form $x^2+c$ 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.</p>}, number={1}, journal={MATHEMATICA SCANDINAVICA}, author={Sadek, Mohammad and Yesin , Tugba}, year={2024}, month={Feb.} }