Almost 2-universal quinary quadratic forms

Abstract

In this thesis, we consider various problems on the representations of binary quadratic forms by positive definite quinary quadratic forms. A (positive definite integral) quadratic form is called almost 2-universal if it represents all (positive definite integral) binary quadratic forms but finitely many up to isometry. We prove that three candidates of almost 2-universal quinary diagonal quadratic forms are indeed almost 2-universal. We also prove that there are at most 16 almost 2-universal quinary quadratic forms having exactly only one exception, and prove the almost 2-universalities of some candidates. A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all (positive definite integral) binary diagonal quadratic forms. We prove that there are at most 34 diagonally 2-universal quinary quadratic forms, and prove the diagonally 2-universalities of 29 candidates among them.