### Abstract:

Two distinct positive integers are friendly if they have the same abundancy index. Numbers that are not friendly are said to be solitary. The class of numbers of the form for primes consists of numbers whose categorization as either friendly or solitary is not known. The numbers 26 and 38 belong to this class. The study of friendly and solitary numbers is important in the sense that it aids in the identification of yet unknown prime numbers. Prime numbers are central in the practice of cryptography, and are of great importance in areas such as computer systems security and generation of pseudorandom numbers. In this thesis, properties of abundancy index function are systematically used to include or discriminate various prime numbers as possible factors of the potential friends of 26 and 38. They are also used to compute the greatest lower bounds for the potential friends of the two numbers. The thesis establishes that a friend of 26 must be of the form where , , and must be larger than . On the other hand, a friend of 38 must be of the form where , , and is not a square, and must be larger than . In addition, it is established that in general, if is a friend of a number of the form , is a prime greater than, where , , , and such that if is a prime factor of for which but , then the power of in must be even.
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CHAPTER ONE
1.0 INTRODUCTION AND LITERATURE REVIEW
1.1 Introduction
Natural numbers are mathematical objects used in counting and measuring, and notational symbols which represent natural numbers are called numerals. The number concept and counting process was being used even in most primitive times in counting elapsed time or keeping records of quantities, such as of animals. Today’s numerals, called Hindu-Arabic numbers are a combination of just 10 symbols or digits: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. In addition to their use in counting and measuring, numerals are often used for numerical representation such as telephone numbers, serial numbers, and construction of codes in cryptography.
Definition 1.1.1 An integer is said to be divisible by a non-zero integer or divides , denoted , if there exists some integer such that . In this case, (and also ) is said to be a factor or divisor of , and is a multiple of (and of ). The notation is used to indicate that is not divisible by .
Definition 1.1.2 The greatest common divisor of two integers and , not both zero, denoted is the poistive integer such that:
and ; and
if and , then c .
Definition 1.1.3 Two integers and , not both zero, are said to be relatively prime or coprime if