The Uniform Boundedness Conjecture is not known even for quadratic polynomials Fc(x) = x2 + c over the rational numbers Q. It is known in this case that Fc(x) cannot have periodic points of period four,[4] five,[5] or six,[6] although the result for period six is contingent on the validity of the conjecture of Birch and Swinnerton-Dyer. Poonen has conjectured that Fc(x) cannot have rational periodic points of any period strictly larger than three.[7]