Numbers whose Difference equals Difference between Cube and Seventh Power

Theorem
The following $2$ pairs of integers are the only ones known which exhibit this pattern:


 * $\left\vert {5^3 - 2^7}\right\vert = 5 - 2$


 * $\left\vert {13^3 - 3^7}\right\vert = 13 - 3$

Proof
We have: