TY - JOUR
AU - Sadek, Mohammad
AU - Yesin , Tugba
PY - 2024/02/26
Y2 - 2024/10/09
TI - A dynamical analogue of a question of Fermat
JF - MATHEMATICA SCANDINAVICA
JA - Math. Scand.
VL - 130
IS - 1
SE - Articles
DO - 10.7146/math.scand.a-142342
UR - https://www.mscand.dk/article/view/142342
SP -
AB - <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>
ER -