Vladimir Arnold defined a perfect form as a particular binary quadratic form F(a,b,c):= ax^2+bxy+cy^2, with all variables integral, such that the product of two such forms is of like form. An easy example can be given by the Brahmagupta-Fibonacci two-square identity which involves x^2+y^2, hence is the case (1,0,1). For what other (a,b,c) does this happen?
We discuss two explicit general identities that can establish certain (a,b,c)'s as a perfect form, with the latter one interestingly connected to discriminants with class number h(d)=3m. A small example would be the form (2,1,3).
Perfect forms can also be generalized to diagonal quaternary quadratic forms (a,b,c,d). These perfect diagonal forms can be given by Ramanujan's 54 quaternary forms, though may not be limited only to these.
http://mathworld.wolfram.com/topics/Piezas.html
