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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
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