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Query: Ternary Quadratic Forms, Modular Equations and Certain Positivity
  Conjectures   We show that many of Ramanujan's modular equations of degree 3 can be
interpreted in terms of integral ternary quadratic forms. This way we establish
that for any n in N
  |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 3y^2 +3z^2:
x,y,zin Z}|, just to mention one among many similar positive results of this
type. In particular, we prove the recent conjecture of H. Yesilyurt and the
first author stating that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in
Z}| >= |{n= x(x+1)/2 + 7y^2 + 7z^2: x,y,z in Z}|. We prove a variety of
identities for certain ternary forms with discriminants 144,400, 784,3600 by
converting these into identities for appropriate eta- quotients. In the process
we discover and prove a few new modular equations of degree 5 and 7. For any
square free odd integer S with prime factorization p_1.....p_r, we define the
S-genus as a union of 2^r specially selected genera of ternary quadratic forms,
all with discriminant 16 S^2. This notion of S-genus arises naturally in the
course of our investigation. It entails an interesting injection from genera of
binary quadratic forms with discriminant -8 S to genera of ternary quadratic
forms with discriminant 16 S^2.

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The categories of this paper are: physics
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