Why are diophantine equations defined as having integer-only
solutions? Is it merely so that those solutions that are integer-only
have a name? It seems easy to find a "diophantine equation" whose
solutions are real or complex, as well as integer. Is it then no
longer a diophantine equation?
I guess I'm trying to get at whether this is just a semantic
distinction or whether there is a deeper reason for the naming
convention. Is there a name for the same equation when referring to
non-integer solutions?