Primitive Representations by Positive Ternary Quadratic Forms: A Combinatorial Approach

Our objective in this paper is to show that if an integral, positive-definite, and primitive ternary quadratic form Q(x,y,z) is alone in its genus and its discriminant is a power of 2, then an explicit formula for the number of primitive representations of a positive integer n by Q, that is Q(x,y,z)=n where gcd(x,y,z)=1, can be obtained and proved in an elementary combinatorial manner. We give the representation numbers of all such ternary quadratic forms in terms of the class number. The proofs are automated via computer support.