  In this article, we study the representability of integers as sums of
pentagonal numbers, where a pentagonal number is an integer of the form
$P_5(x)=\frac{3x^2-x}{2}$ for some non-negative integer $x$. 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.
