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The purpose of this note is give some evidence in support of conjectures ofPoonen, and Morton and Silverman, on the periods of rational numbers under theiteration of quadratic polynomials. In particular, Poonen conjectured thatthere are at most 9 periodic points defined over the rational numbers for anymap in the family x^2 + c for c rational. We verify this conjecture for cvalues up to height 10^8. For quadratic number fields, we provide evidence thatthe upper bound on the exact period of Q-rational periodic point is 6.

On Poonen's Conjecture Concerning Rational Preperiodic Points of Quadratic Maps