#### Abstract

For subsets of natural numbers S and W, we say S forces W if any integer-matrix positive definite form which represents every element of S over the integers also represents every element of W over the integers. In the context of a superset S*, S is referred to as a unique minimal forcing set of W if for any subset S0 of S*, we have that S0 forces W if and only if S is a subset of S0. In 2000, Manjul Bhargava used his own novel method of “escalators" to prove the unique minimal forcing set of the natural numbers is T={1, 2, 3, 5, 6, 7, 10, 14, 15}, which was a refinement of the celebrated Conway-Schneeberger Fifteen Theorem. We use Bhargava's theory of escalators to develop an algorithm which determines whether a positive integer, interpreted as a singleton in the natural numbers, has a unique minimal forcing set within T and to establish several infinite families of positive integers without unique minimal forcing sets in T.

#### Sponsor

Andrew Obus, Department of Mathematics, University of Virginia

#### Recommended Citation

Rabbani, Tahseen
(2016)
"Unique minimal forcing sets and forced representation of integers by quadratic forms,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 17
:
Iss.
1
, Article 1.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol17/iss1/1