Integrality of Open Instantons Numbers

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.