  For $c \in \mathbb{Q}$, consider the quadratic polynomial map
$\varphi_c(x)=x^2-c$. Flynn, Poonen and Schaefer conjectured in 1997 that no
rational cycle of $\varphi_c$ under iteration has length more than $3$. Here we
discuss this conjecture using arithmetic and combinatorial means, leading to
three main results. First, we show that if $\varphi_c$ admits a rational cycle
of length $n \ge 3$, then the denominator of $c$ must be divisible by $16$. We
then provide an upper bound on the number of periodic rational points of
$\varphi_c$ in terms of the number of distinct prime factors of the denominator
of $c$. Finally, we show that the Flynn-Poonen-Schaefer conjecture holds for
$\varphi_c$ if that denominator has at most two distinct prime factors.
