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This thesis is presented as an introduction to the Fibonacci sequence of integers. It is hoped that this thesis will create in the reader more interest in this type of sequence and especially the Fibonacci sequence. It seems that this particular area of mathematics is often ignored in the classroom or touched upon far too briefly to stimulate curiosity and develop further interest in this field.

The primary purpose of this thesis is to give a substantial generalization of the set of integers Z, where particular emphasis is given to number theoretic questions such as that of unique factorization. The origin of the thesis came from a study of a special case of generalized integers called the Gaussian Integers, namely the set of all complex numbers in the form n + mi, for m,n in Z. The main generalization involves what are called algebraic integers.

Let D be an integral domain and f a polynomial that is integer-valued on D. We prove that Int(f(D);D) has the Skolem Property and give a description of its spectrum. For certain discrete valuation domains we give a basis for the ring of integer-valued even polynomials. For these discrete valuation domains, we also give a series expansion of continuous integer-valued functions. / by Mario V. Marshall. / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.

abstract: 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. / Dissertation/Thesis / M.A. Mathematics 2010

Over the last decade, the demand for privacy and data confidentiality in communication and storage processes have increased exponentially. Cryptography can be the solution for this demand. However, the critical issue occurs when there is a need for computing publicly on sensitive information or delegating computation to untrusted machines. This must be done in such a way that preserves the information privacy and accessibility. For this reason, we need an encryption algorithm that allows computation on information without revealing details about them. In 1978 Rivest, Adleman and Dertouzos raised a crucial question: can we use a special privacy homomorphism to encrypt the data and do an unlimited computations on it while it remains encrypted without the necessity of decrypting it? Researchers made extensive efforts to achieve such encryption algorithm. In this paper, we introduce the implementation of the CRT-based somewhat homomorphic encryption over the integers scheme. The main goal is to provide a proof of concept of this new and promising encryption algorithm. / Graduate

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [52]-53).

This paper will be devoted to an examination of several general and specific equations and systems of equations of the diophantine type. Only algebraic equations with integral coefficients, not all zero, will considered. The elementary properties of the integers will be assumed.

The purpose of this paper is to construct the real number system. The foundation upon which the real number system will be constructed will be the system of counting numbers.

Magister Educationis - MEd / This research investigated how grade 7 learners dealt with introductory aspects of integers when they are introduced through a temperature model. In particular, the study analysed the effect of an intentional teaching strategy on learners’ engagement with integers. The idea of combining an intentional teaching strategy with the introduction of integers in grade 7 learners using a temperature model is what makes this study unique. A qualitative study was adopted. Data was collected by means of audio and video and also by means of learners’ completed worksheets. The results of the study indicate that the majority of learners could recognise, compare and order integers. It is recommended that the application of intentional teaching with a temperature model is a viable strategy to introduce grade 7 learners to integers.