Elementary number theory with programming is a useful textbook for undergraduate and graduatelevel students majoring in mathematics or computer science, as well as an excellent supplement for teachers and students who would like to better understand and appreciate number theory and computer programming. Each worksheet is randomly generated and thus unique. 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. Second edition victor shoup frontmatter more information. Number theory is one of the oldest and most beautiful branches of mathematics. These notes were prepared by joseph lee, a student in the class, in collaboration with prof. Cryptography hash functions iii however, the function is clearly not onetoone. In this section we will describe a few typical number theoretic problems. Word problems involving greatest common factor and least. Note that these problems are simple to state just because a topic is accessibile does not mean that it is easy. Waclaw sierpinski 250 problems in elementary number theory presents problems and their solutions in five specific areas of this branch of mathe matics. Balloons are sold in bags of 20, party horns are sold in bags of 10, and there are 8 candy bars in a package.
Number theory is replete with sophisticated and famous open problems. In the problem for n 304 24 34 54, a divisor of n with an odd number of divisors must have 0, 2, or 4 factors of. This textbook takes a problem solving approach to number theory, situating each theoretical concept within the framework of some examples or some problems for readers. Some typical number theoretic questions the main goal of number theory is to discover interesting and unexpected relationships between different sorts of numbers and to prove that these relationships are true. Example 1 the number 102 has the positive divisors 1, 2, 3, 6, 17, 34, 51, 102, and the number 170 has the positive divisors 1, 2, 5, 10, 17, 34, 85, and 170. All considerations of the next two sections carry over word for word in. Number theory and algebra play an increasingly signi.
Mordells proof of the three squares theorem 101 15. This book teaches number theory through problem solving and is designed to be selfstudy guide or supplementary textbook for a onesemester course in. Theory presents problems and their solutions in five specific. Selected number theory exercises university of notre dame. You will also see problems involving prealgebra concepts such as negative numbers, exponents, and square roots getting you ready for the algebra in section 5. Basic number theory1 practice problems math hackerearth. This book was written as a text for the learning of number theory, not as. Other things you can try in tackling a problem include changing the. If we compute mod n on both sizes, nqc cancels out and we obtain ac bc mod n. The answer key is automatically generated and is placed on the second page of the file you can use the generator to make worksheets either in html or pdf format both are easy to print. Some numbertheoretic problems that are yet unsolved are. The exposition is informal, with a wealth of numerical examples that are analyzed for patterns and used to make conjectures. If youre looking for a historical presentation of number theory, john stillwells elements of number theory cant be beat. A computational introduction to number theory and algebra version 2 victor shoup.
In this book, all numbers are integers, unless specified otherwise. Aside from an otherwise standard flow of theorems, proofs, exercises, etc, there are approximately 800 problems all with solutions. An example is checking whether universal product codes upc or international standard book number isbn codes are legitimate. Pdf 100 number theory problems with solutions amir. His discrete math textbook is the standard book for math 55.
Word problems on average speed word problems on sum of the angles of a triangle is 180 degree. Analytic number theory a tribute to gauss and dirichlet 7 ams cmi duke and tschinkel, editors 264 pages on 50 lb stock 12 inch spine analytic number theory a tribute to gauss and dirichlet william duke yuri tschinkel editors cmip7. To determine the greatest common divisor by nding all common divisors is. This section will provide a war mup session before you move on to more dif. A friendly introduction to number theory is an introductory undergraduate text designed to entice nonmath majors into learning some mathematics, while at the same time teaching them how to think mathematically. 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. Jul 11, 2007 the heart of mathematics is its problems. It covers the basic background material that an imo student should be familiar with. Thus arose the problem of prime numbers and the other problems of number theory, galoiss theory of equations, the theory of algebraic invariants, the theory of abelian and automorphic functions. Magellan has decided to make party baskets for the fund raiser. Estimates of some functions on primes and stirlings formula 15 part 1. There is, in addition, a section of miscellaneous problems. Its a bit more difficult then the other 2 books as it functions as a companion to his abstract algebra text and presents the integers in the context of the development of basic ring theory from the ancient greek to today.
Applications cse235 introduction hash functions pseudorandom numbers representation of integers euclids algorithm c. 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. God made the integers, all else is the work of man. Any operation add, subtract, multiply, divide, exponent with polynomials. Basic algorithms in number theory universiteit leiden. Number theory naoki sato 0 preface this set of notes on number theory was originally written in 1995 for students at the imo level. The problem was solved in a few months by wallis and brouncker. Khatri, an interesting geometrical progression, scripta math. In a book he was reading around 1630, fermat claimed to have a proof, but not enough space in the margin to write it down. Also go through detailed tutorials to improve your understanding to the topic. Practice number theory problems 3 b show that if a b mod n, then for all positive integers c, ac bc mod n. Word problems involving highest common factor and lowest common multiple read each question carefully and think about what the question is asking. For example, here are some problems in number theory that remain unsolved. The largest number allowed for gcf is 1,000,000 and for lcm 10,000 to be safe to reduce the server load.
What are some good number theory books for beginners. When two elements, x 1 6 x 2 hash to the same value, we call it a collision. Solve practice problems for basic number theory1 to test your programming skills. How many of each should he buy so there are an equal number of balloons, horns and candy bars in each basket. We prove dirichlets theorem which says that every irrational number can be approximated by in. This textbook takes a problemsolving approach to number theory, situating each theoretical concept within the framework of some examples or some problems for readers.
Paul halmos number theory is a beautiful branch of mathematics. Have protractor, ruler and compass ready to measure angles and lengths. Divisibility is an extremely fundamental concept in number theory, and has applications including puzzles, encrypting messages, computer security, and many algorithms. Some applications of the three squares theorem 103 15. To find all of the prime numbers between 1 and 100, complete the following exercise. One of the most common hash functions uses modular arithmetic. Release notes this version of yaintt has a particular emphasis on connections to cryptology. Adler and courys text on elementary number theory is one of the best ive ever seen. The purpose of this book is to present a collection of interesting problems in elementary number theory.
My goal in writing this book was to provide an introduction to number theory and algebra, with an emphasis. Number theory is a branch of pure mathematics devoted primarily to the study of the integers. The word arithmetic is used by the general public to mean elementary. The riemann hypothesis, which is the most famous unsolved problem in number theory, postulates a very precise answer to the question of how the prime numbers are distributed. Many of the problems are mathematical competition problems from all over the world like imo, apmo, apmc, putnam and many others. Find the prime factors for each number draw a prime factor diagram remember numbers in the overlap hcf numbers in the whole diagram lcm 1.
Combinatorial and analytic number theory mathematical institute. All of the mathematics required beyond basic calculus is developed from scratch. It abounds in problems that yet simple to state, are very hard to solve. Goldbachs conjecture is every even integer greater than 2 the sum of distinct primes. Selected number theory exercises exercise 1 suppose n1 is an integer such that 4n 1. Problems are taken from imo, imo shortlistlonglist, and some other famous math competitions.
Whereas one of the principal concerns of the latter theory is the deconposition of numbers into prime factors, additive number theory deals with the decomposition of numbers into summands. Given two integers aand bwe say adivides bif there is an integer csuch that b ac. In the school year 195960, these seventh and eighth grade books were used by about 175 teachers in many parts of the country,and then further revised in the summer of 1960. We have shown that every positive integer can be written as the sum four squares.
Number theory has to do with the study of whole numbers and their special properties. Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself. Basic algorithms in number theory 27 the size of an integer x is o. Elementary number theory and its applications by kenneth h. A huge chunk of number theory problems are diophantine equations named after an ancient greek math ematician diophantus. Self study of number theory mathematics stack exchange. Free worksheets for the greatest common factor gfc and. The topics of this chapter belong to a branch of mathematics known as number theory. I remember liking his number theory book when it first came out, but i havent seen a copy lately.