In this talk, we study the representability of integers as sums of pentagonal numbers. In particular, we prove the "pentagonal theorem of 63", which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers 1, 2, 3, 4, 6, 7, 8, 9, 11, 13, 14, 17, 18, 19, 23, 28, 31, 33, 34, 39, 42, and 63.

We also introduce a method to obtain a generalized version of Cauchy's lemma using representations of binary integral quadratic forms by quaternary quadratic forms, which plays a crucial role in proving the results. This is a joint work with Daejun Kim.