Re why would we care: this is a search algorithm for numerical coincidences. Most numerical coincidences are trivial, for example can be derived from hypergeometric function relation which was known to Gauss. In fact it would be interesting to automatically filter formulae which can be derived from hypergeometric function relation... On the other hand, numerical coincidences can lead to deep theory, monstrous moonshine is a prime example. Hope is that by searching for numerical coincidences, we can discover one leading to deep theory without already knowing that deep theory. This seems reasonable.
Table 3 inside also shows new conjectures for constants such as Catalan's and zeta(3).
These results do not seem to trivially arise from known knowledge.
