Although I didn't know offhand what the equivalence classes of binary quadratic forms over Z is, I still meant to point out that: ----- This could be used to simplify the process of determining which numbers are represented by a given quadratic form f. If say f is equivalent over Z to the standard representative g_k, then it represents the same integers as g_k, which can be known in advance. ----- where "this" means: a) the presumed classification, plus b) knowing what integers each equivalence class represents, plus c) how to determine, in terms of (a,b,c), which equivalence class the quadratic form a x^2 + b x y + y^2 belongs to. My comment is just that once those things are established they might (emphasis on might) provide a shortcut to determining the set of integers that the given quadratic form represents. —Dan