  We prove the integrality of the open instanton numbers in two examples of
counting holomorphic disks on local Calabi-Yau threefolds: the resolved
conifold and the degenerate $ \P \times \P $. Given the B-model superpotential,
we extract by hand all Gromow-Witten invariants in the expansion of the A-model
superpotential. The proof of their integrality relies on enticing congruences
of binomial coefficients modulo powers of a prime. We also derive an expression
for the factorial $(p^k-1)!$ modulo powers of the prime $p$. We generalise two
theorems of elementary number theory, by Wolstenholme and by Wilson.
