Discipline: Mathematics and Statistics
Chanae Ottley - University of the Virgin Islands
Co-Author(s): Chris Plyley, University of the Virgin Islands, U.S. Virgin Islands
Given a finite additive group G, the study of sequences of (not necessarily distinct) elements of G which sum to zero has numerous applications in game theory, cryptography, Ramsey theory and graph theory. For instance, the determination of Davenport’s Constant, which is defined as the smallest integer k such that any sequence of k elements has a zero-sum subsequence, is considered one of the most important unsolved problems in all of finite group theory. When a sequence is assumed to have no repeated elements, the analog is Olson’s Constant (denoted Ol(G)), which is defined as the smallest integer k such that every sequence of k distinct elements has a zero-sum subsequence. The determination of Ol(Zn) is an open problem, and even for Zn (the group of integers under addition modulo n) the precise values of Ol(Zn) are only known when n<65. In 1988, D. Kleitman and P. Lemke () introduced the index of a sequence while resolving a conjecture of Paul Erdös. If S is a zero-sum sequence in Zn, then the index of S is the minimal integer multiple of n that S may be made to sum to after multiplication by an integer coprime to n. In 2011, Gao et al. () formally asked: what is the smallest integer k such that every sequence of length k has a zero-sum subsequence with index 1? In this project, we determine this constant, denoted by T(n), for all n<25. To accomplish this, we first establish upper and lower bounds on the value of T(n). Beginning with the lower bound, we use a computer program to generate a list of all possible sequences of this length, and then utilize a second computer program to search these sequences for zero-sum sequences with index 1. Any remaining sequences are checked by hand until a value of T(n) is determined. Our data suggests that the value of T(n) remains relatively close to the value of Ol(Zn) as n increases. This supports the idea that sequences with no subsequences of index 1 are relatively sparse. Future research projects include: determining the value of T(n) for all n<65 (which will require high powered computer programs), formulating a function which expresses the value T(n) in terms of Ol(Zn), and extending our ideas to include either more general groups or sequences with repetition. References:  W. Gao, Y. Li, J. Peng, C. Plyley, and G. Wang, On the index of sequences in cyclic groups. Acta Arithmetica, 148(2) (2011), 119-134.  D.J. Kleitman and P. Lemke, An addition theorem on the integers modulo n. J. Number Theory, 31 (1989), 335-345.
Funder Acknowledgement(s): National Science Foundation/ HBCU-UP.
Faculty Advisor: Chris Plyley, firstname.lastname@example.org
Role: For this research project, I worked on determining what value of T(n) would allow sequences with length n to be zero-sum sequences of index 1. After generating the list of sequences and running it through a computer program, the remaining sequences were checked by hand. Those sequences were multiplied by a number coprime to n to establish if they are sequences with index 1. This process was continued for every remaining sequence until the value of T(n) was determined for all n less than 25.