## Theses, Dissertations and Culminating Projects

5-2011

Thesis

#### Degree Name

Master of Science (MS)

#### College/School

College of Science and Mathematics

#### Department/Program

Mathematical Sciences

Aihua Li

Michael A. Jones

Jonathan Cutler

#### Abstract

In this paper, I investigate polynomial solutions to the Diophantine equa­ tion, X² +Y³ = 6912Z², where X = g(x,y), Y = h(x,y) and Z = f(x,y) are polynomials with integer coefficients. The focus is on the greatest common di­ visors for the integer values of these polynomials when the polynomials f (x, y), g(x, y) and h(x, y) are relatively prime in Q[x, y]. However, for a fixed integer pair xo, Yo, the integer values f(x0,y0), g(x0, y0) and h(x0,y0) are not necessarily relatively prime in Z. I investigate the greatest common divisors (GCDs) of these three polynomial values for specific integer pairs x0 and y0· First, I study the cases where y0 = 1 and y0 = 2. For these cases, a complete distri­bution of the GCDs is given. Furthermore, I use the Euclidean Algorithm and Grobner Basis techniques to determine the GCDs for f(x0, y0), g(x0, y0) and h(x0, y0) in Z by obtaining multiples of the GCDs of the polynomials. Then, the results from the cases y0 = 1 and y0 = 2 are generalized to obtain similar properties of the GCDs for all possible integer values of x and y. For the cases where the integer values are not relatively prime, the possible prime divisors of the GCDs and integer bounds for the powers of prime divisors are determined. Finally, polynomial solutions to new Diophantine equations are derived from the original Diophantine equation.

PDF

COinS