Suppose we define a set: $$ S := \{(x,y) \in \mathbb{N}\times \mathbb{N} | x^y =y \mathbin\Vert x\} $$ where "$\mathbin\Vert$" denotes the concatenation operation.
We have the example of the pair of number $(5,2) \in S$. This is because $5^2 = 25 = 2 \mathbin\Vert 5$.
My question is, does there exist a name for these types of numbers or a similar definition (say the numbers belong in a more general set than $ \mathbb{N}$, for example)? And if so, is there any documented research done with similar types of numbers?