The mathematical material covered includes the basics of number theory including unique factorization, congruences, the distribution of primes, and quadratic reciprocity and of abstract algebra including groups, rings. Exploring computational number theory part 1 codeproject. However, it was noticed by chevalley and weil that the situation was improved somewhat if the number. Introduction to the interface of probability and algorithms aldous, david and steele, j. Explore the sequences of quadratic residues modulo m for numbers of the form mp n for p a fixed prime. For your convenience, i have digested the below information into a timeline. Final project 3 goldsteins analytic number theory, tates thesis in casselsfrohlich, gtm book by ramakrishnan and valenza, neukirch. In our brief survey of some recent developments in number theory, we will. The websites by chris caldwell 2 and by eric weisstein are especially good. It contains problems of different levels in number theory and is a good source for practicing regional and international olympiads. Cryptography hash functions ii in general, a hash function should have the following properties it must be easily computable. The game is called sprouts and it is an invention of john horton conway. Elementary number theory projects there are going to be six di erent projects. Pdf some recent developments in number theory researchgate.
Algebraic number theory involves using techniques from mostly commutative algebra and nite group theory to gain a deeper understanding of the arithmetic of number elds and related objects e. Moreover, these notes shed light on analytic number theory, a subject that is rarely seen or approached by undergraduate students. Transcendental number theory a course by kannan soundararajan latexed by ian petrow september 19, 2011 contents 1 introduction. An introduction to the theory of numbers fifth edition. Elementary number theory projects there are going to be. This embedding is very useful in the proofs of several fundamental theorems. This book presents material suitable for an undergraduate course in elementary number theory from a computational perspective. Mathematics archives topics in mathematics number theory. I will also post a pdf of the two documents to the website. Tijdeman, an extension of a theorem of euler, acta arith.
Math 215 project number 1 graph theory and the game. This game is similar to 20 questions and can be adjusted to students level of understanding of numbers and number theory. Applications cse235 introduction hash functions pseudorandom numbers representation of integers euclids algorithm c. Five of the projects will have three participants and the sixth will have only two. The journal of number theory jnt features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. To see what is going on at the frontier of the subject, you may take a look at some recent issues of the journal of number theory which you will. He proved the fundamental theorems of abelian class. In other words, there is a nonzero integral vector xsuch that q x 0. The mission of the journal is to publish highquality original articles that make a significant contribution to these research areas. The journal of number theory features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and modular arithmetic, but also to develop their ability to formulate and test precise conjectures from experimental data. It covers the basic background material that an imo student should be familiar with.
What are some ideas for a research projectpaper on number. An irrational number is a number which cannot be expressed as the ratio of two integers. Number theory is an important part of middle level mathematics and sets the stage for high school, college, and even future careers. Indeed, in a much quoted dictum, he asserted that mathe matics is the queen of the sciences and the theory of numbers is the queen of mathematics. German mathematician carl friedrich gauss 17771855 said, mathematics is the queen of the sciencesand number theory is the queen of mathematics. The ideals that are listed in example 4 are all generated by a single number g. Some of his famous problems were on number theory, and have also been in. We will encounter all these types of numbers, and many others, in our excursion through the theory of numbers.
However, we also consider questions of other types. Introduction to number theory final project first details. It provides students an opportunity to think about the many ways numbers can be described oddeven, whole number, fraction less than or greater than one, prime number, multiple of n, etc. A computational introduction to number theory and algebra. According to cal the largest known prime as of march 2007 is the 44th known mersenne prime p 232582657 1. These notes were prepared by joseph lee, a student in the class, in collaboration with prof. You are free to choose which problems in this project that you would like to work on, all i ask is that you work on things which you havent seen before, and that you do not use any books or the internet. The exercises are carefully chosen to broaden the understanding of the concepts. An introduction to elementary set theory guram bezhanishvili and eachan landreth 1 introduction in this project we will learn elementary set theory from the original historical sources by two key gures in the development of set theory, georg cantor 18451918 and richard dedekind 18311916. Number theory is the study of the set of positive whole numbers. In the course of the problems we shall also work on writing proofs that use mathematical. Estimates of some functions on primes and stirlings formula 15 part 1. A valuable resource for mathematicians, the journal provides an international forum for.
Number theory is an attractive way to combine deep mathematics with fa. The students in my number theory courses at the university of maine have helped with this project. Pdf two topics in number theory a formula for primes and an. Introduction to number theory and its applications lucia moura winter 2010 \mathematics is the queen of sciences and the theory of numbers is the queen of mathematics. Im not that well versed in number theory, but since you mention computer sc. Unless you choose to take a final exam, you will need to do a final project.
This article explores computational number theory and the relationships between the various classical number theorists theories by using euler pseudoprimes to bridge these theories in a base 2 computational environment while providing the user the information in a human readable form to enhance understanding in an exploratory environment. That said, you can still do interesting problems and write them up. Shorey, grimms conjecture on consecutive integers, int. Olympiad number theory through challenging problems. Math 215 project number 1 graph theory and the game of sprouts this project introduces you to some aspects of graph theory via a game played by drawing graphs on a sheet of paper. Can you find any patterns to the number of quadratic residues or the distribution of the residues themselves. This is explained in laboratory 12, in the documentation of the program, and also in the demonstration provided by the program inddem. A rational number is a number which can be expressed as the ratio a b of two integers a,b, where b 6 0. You may copy it, give it away or reuse it under the terms of the project gutenberg license included with this ebook or online at. This project is about several different way of expressing a real num ber. Students were asked to choose a serious topic in number theory of which the projects were my suggestions with useful references to write a final paper of at least 10 pages explaining their topic i.
Adobe pdf, postscript and dvi readers the fibonacci numbers add. An example is checking whether universal product codes upc or international standard book number isbn codes are legitimate. A2a as a high schooler, you likely dont have the prerequisites to do actual mathematical research. Project gutenbergs essays on the theory of numbers, by.
We next show that all ideals of z have this property. Preface these are the notes of the course mth6128, number theory, which i taught at queen mary, university of london, in the spring semester of 2009. Students will apply a variety of mathematical concepts and skills to. Ram murty and others published some recent developments in number theory find, read and cite all the research you need on researchgate. It is nothing but a set of problems posted by active users of aopsmathlinks. This chapter lays the foundations for our study of the theory of numbers by weaving together the themes of prime numbers, integer factorization, and the distribution of primes. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field. It should distribute items as evenly as possible among all values addresses. Project interactivate activities number and operation concepts add. Gauss and number theory without doubt the theory of numbers was gauss favourite sub ject. Divisibility is an extremely fundamental concept in number theory, and has applications including puzzles, encrypting messages, computer security, and many algorithms.
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