Document Type

Thesis - Open Access

Award Date


Degree Name

Master of Science (MS)


Mathematics and Statistics


One of the most interesting and commonly pursued branches of mathematics is the study of number theory. Its appeal lies not in its applicability, but in a fascination for the properties of the numbers themselves. It should be noted that, on occasion, problems of number theory have contributed to more pragmatic branches of mathematics and hence to a practical application. Frequently this happens without the original author's expectation or intention since he was merely remarking on what he considered to be a singular or unusual facet of the employed number system. Such a problem is that which involves the rearrangement of the digits of a given integer. The algebra student frequently encounters problems of this nature while studying systems of equations. He is asked to find an integer such that the sum of this integer and the integer generated by reversing the order of the digits is equal to a second given integer. The student, if he progresses to the study of an actual course in number theory, will again encounter problems involving reversing the order of the; digits of unknown integers, and at this time further stipulations will be added. It is this problem with which this paper will deal. In this paper, as in most work done in number theory, we shall be using the set of integers and any alphameric symbols, unless stated otherwise, will be understood to represent integers. We shall employ the system of axioms associated with the integers; this system can be found in any elementary algebra text. In addition, we shall here introduce a further postulate which will be employed in one of the following three equivalent forms: (1) First form of the Principle of Mathematical Induction. Any set of positive integers which contains the integer 1 and the integer k+l whenever it contains the positive integer k, contains all positive integers. (2) Second form of the Principle of Mathematical Induction. Any set of positive integers which contains 1 and k+l whenever it contains the integers 1 to k inclusive, contains all positive integers. (3) Well Ordering Principle. Every non-empty set of positive integers has a least element.

Library of Congress Subject Headings

Number theory



Number of Pages



South Dakota State University