Binary quadratic forms representing same multiples of a prime

Abstract

For a positive definite binary quadratic form f(x, y) = ax^2+bxy+cy^2 for a, b, c in Z, the set of integers that are represented by f is denoted by Q(f). In this thesis, we find all pairs (f, g) of binary quadratic forms such that Q(f) \cap pZ = Q(g) \cap pZ, where p = 2, 3 or 5 and the binary quadratic forms f_p(x, y), g_p(x, y) defined over Z_p are isotropic 1/2Z_p-modular. In Chapter 4, we use the method developed in [4] to find all such pairs.