  In this paper, we will apply the ideas from the mirror symmetry of Calabi-Yau
threefolds to study the modular forms and one-parameter family of K3 surfaces
found by Beukers and Peters, which provide enlightenment to the two mysterious
sequences constructed by Ap\'ery in his proof of the irrationality of
$\zeta(3)$. We will construct a fourth order differential operator and a
prepotential from the canonical solutions of this differential operator. The
third derivative of this prepotential with respect to the mirror map defines a
Yukawa coupling that is a weight-4 modular form. The instanton expansion of
this Yukawa coupling yields integral instanton numbers, which are also periodic
with period 6.
