Description
ABSTRACT This thesis attempts to answer two questions based upon the historical observation that 1^2 +2^2 +· · ·+24^2 = 70^2. The first question considers changing the starting number of the left hand side of the equation from 1 to any perfect square in the range 1 to 10000. On this question, I attempt to determine which perfect square to end the left hand side of the equation with so that the right hand side of the equation is a perfect square. Mathematically, Question #1 can be written as follows: Given a positive integer r with 1 less than or equal to r less than or equal to 100, find all nontrivial solutions (N,M), if any, of r^2+(r+1)^2+···+N^2 =M^2 with N,M elements of Z+. The second question considers changing the number of terms on the left hand side of the equation to any fixed whole number in the range 1 to 100. On this question, I attempt to determine which perfect square to start the left hand side of the equation with so that the right hand side of the equation is a perfect square. Mathematically, Question #2 can be written as follows: Given a positive integer r with 1 less than or equal to r less than or equal to 100, find all solutions (u, v), if any, of u^2 +(u+1)^2 +(u+2)^2 +···+(u+r-1)^2 =v^2 with u,v elements of Z+. The two questions addressed by this thesis have been on the minds of many mathematicians for over 100 years. As a result of their efforts to obtain answers to these questions, a lot of mathematics has been developed. This research was done to organize that mathematics into one easily accessible place. My findings on Question #1 can hopefully be used by future mathematicians in order to completely answer Question #1. In addition, my findings on Question #2 can hopefully be used by future mathematicians as they attempt to answer Question #2 for values of r greater than 100.
Details
Contributors
Roth, Sanford Gary (Author)
Bremner, Andrew (Thesis advisor)
Childress, Nancy E (Committee member)
Jones, John W. (Committee member)
Arizona State University (Publisher)
Date Created
The date the item was original created (prior to any relationship with the ASU Digital Repositories.)
2010
Topical Subject
Resource Type
Language
eng
Note
thesis
Partial requirement for: M.A., Arizona State University, 2010
bibliography
Includes bibliographical references (p. 53)
Field of study: Mathematics
Citation and reuse
Statement of Responsibility
by Sanford Gary Roth
Additional Information
Extent
iii, 53 p. : ill