Representations by a binary quadratic form with class number 3

Abstract

For integers a, b, c, the homogeneous quadratic polynomial f(x,y)=ax^2+bxy+cy^2 is called a binary quadratic form. In this thesis, we will give a closed formula on the number of integral solutions of x^2+27y^2=n for any integer n. To do this, we follow the framework given by Min and Oh, where they considered the binary form x^2+32y^2. In section 2, we briefly survey on the theory of binary quadratic forms and give some lemmas which are needed in the later. In section 3, we consider the case when n is a prime power. In section 4, we consider the case when n is any integer relatively prime to 6. In section 5, we summarize everything and give a complete formula on the number of solutions of the above equation.