## Mathematics & Computer Science

#### Event Title

Perfect Difference Sets

#### Document Type

Oral Presentation

#### Location

Indianapolis, IN

#### Start Date

13-4-2018 2:00 PM

#### End Date

13-4-2018 2:45 PM

#### Sponsor

Ankur Gupta (Butler University), Jonathan Webster (Butler University)

#### Description

Perfect difference sets are a set of residues, or remainders, under the modulo difference operation. This set, S, contains n elements drawn from V = {0, 1, 2, . . ., v-1}, where v is of the form n^2+ n + 1. Per the Prime Power Conjecture, these sets only exist when n is a prime power. All nonzero residues in V can be expressed uniquely in the form x - y (mod v) for x and y in S. The existence of perfect difference sets has been verified for n < 2,000,000,000 by L. Baumert and D. Gordon.

We implemented and analyzed tests for perfect difference sets developed by T. Evans and H. Mann in a computer program. In particular, we reorganized the tests according their run time and eliminated numbers more quickly. Using this technique, we also verified the Prime Power Conjecture up to n < 1,000,000,000, but we encountered storage constraints. By restructuring the problem in residue classes, we are now able to reduce the storage complexity. We suspect it may also dramatically improve time efficiency. In fact, we are redesigning our implementation to leverage residues and verify n up to 10^14.

Perfect Difference Sets

Indianapolis, IN

Perfect difference sets are a set of residues, or remainders, under the modulo difference operation. This set, S, contains n elements drawn from V = {0, 1, 2, . . ., v-1}, where v is of the form n^2+ n + 1. Per the Prime Power Conjecture, these sets only exist when n is a prime power. All nonzero residues in V can be expressed uniquely in the form x - y (mod v) for x and y in S. The existence of perfect difference sets has been verified for n < 2,000,000,000 by L. Baumert and D. Gordon.

We implemented and analyzed tests for perfect difference sets developed by T. Evans and H. Mann in a computer program. In particular, we reorganized the tests according their run time and eliminated numbers more quickly. Using this technique, we also verified the Prime Power Conjecture up to n < 1,000,000,000, but we encountered storage constraints. By restructuring the problem in residue classes, we are now able to reduce the storage complexity. We suspect it may also dramatically improve time efficiency. In fact, we are redesigning our implementation to leverage residues and verify n up to 10^14.