For
$c \in \mathbb {Q}$
, consider the quadratic polynomial map
$\varphi _c(z)=z^2-c$
. 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 s of distinct prime factors of the denominator of c. Finally, we show that the Flynn–Poonen–Schaefer conjecture holds for
$\varphi _c$
if
$s \le 2$
, i.e., if the denominator of c has at most two distinct prime factors.