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
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thesisPartial requirement for: M.A., Arizona State University, 2010
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bibliographyIncludes bibliographical references (p. 53)
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Field of study: Mathematics
Citation and reuse
Statement of Responsibility
by Sanford Gary Roth
Additional Information
Extent
- iii, 53 p. : ill