Universal Quadratic Forms and the 15-Theorem and 290-Theorem

We study the problem of universal quadratic forms, whose solution is given by the recent paper of Bhargava and Hanke on the 15-Theorem and 290-Theorem. The main purpose of this paper is to explicate Bhargava and Hanke’s proofs of the 15-Theorem and 290-Theorem, basically following their methods throughout the paper. We provide some details omitted in their original proof, and work out several examples to illustrate the idea clearly