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Rational periodic points of the quadratic function $$Q_ c(x)=x^ 2+c$$. (English) Zbl 0804.58036
A condition for the existence of rational fixed points and periodic points (of periods 2 and 3) for $$Q_ c(x) = x^ 2 + c$$ depending on $$c$$ is given (also a condition for nonexistence of rational periodic points of higher periods). There is a connection to Pythagorean triples and some results on periodic points of $$Q_ c$$ treated as a function over $$p$$-adic numbers.

##### MSC:
 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 11R09 Polynomials (irreducibility, etc.) 37P45 Families and moduli spaces in arithmetic and non-Archimedean dynamical systems
##### Keywords:
quadratic function; $$p$$-adic numbers; periodic points
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