Number Theory

Math. 301, Sec. 001, Fall 2017

Homework and Announcements

Urgent stuff

Please NOTE! All handouts for the class will be posted as pdf files after a bullet sign. All these handouts are copyright material and are intended for the students of this class only. IN PARTICULAR, DO NOT POST THESE HANDOUTS ON ANY OTHER WESITE OR INTO AN ARCHIVE DATABASE!

GMU Undergraduate Math competitions Activities Two upcoming competitions are: (i) Virginia Tech Math Competition Saturday, October 21, 2017, 9 -- 11:30 am. (ii) Putnam Math Competition Saturday, December 2, 2017, 10 am -- 1 pm (Part A) and 3 pm -- 6 pm (Part B).

Those who are interested should contact either Prof. Dan Anderson (danders1"the-at-sign"gmu-dot-edu), or Prof. Jim Lawrence (lawrence"the-at-sign"gmu-dot-edu) for more info. -- Recruitment is underway, and practice for the above competitions will be: Friday Putnam Practice (aka "Chill Math") most Friday's at 4:45 -- 6 pm at Math Dept. Room 4307 (open collaboration room) EXPLORATORY HALL.

General Info

For general information regarding this course, please read the Syllabus here below, it should answer all practical questions, like when tests are, what the grading policies are etc.

Homework and Solutions

Here below the homework (HW) assignments and selected solutions will be posted. Many of the problems will be exercises from our textbook. -- Please be sure to read the solutions. If you have questions or comments feel free to come and talk to me.

There is no HW for Labor Day Monday, since there is no class.

  1. Homework for Monday, Sept. 11. : Be sure that you know how inductive proofs work. Every form of induction (both weak and strong form) is a consequence of the Well Ordering Property (WOP) of the natural numbers: Every nonempty set of natural numbers has a least element. When defining a sequence inductively, we call it recursion . To prove some property for a recursively defined sequence it is very often good to use induction.
  2. Homework for Monday, Sept. 18 (I assigned some problems too eagerly, sorry.!) Countable sets are precisely those sets where we can list their elements in the same manner as the natural numbers: the first element, the second element and so on. Many familiar sets are countable, but not all. (We will see that the set of real numbers is, in fact, not countable.) -- At this point (the end of the "Cook's Tour") we know that there are many number sets out there. Some are countable, meaning that we can make a list of them number 1, 2, 3 and so on, that will cover the whole set (just like the natural numbers), and some sets of numbers are uncountable, like the set of real numbers and the complex numbers. This means that there is no way one can write a computer code to list all the real numbers say. -- The integers are interesting (and also difficult for that matter!) since we cannot always divide one number with another nonzero number and obtain an integer. When solving equations involving integers we must proceed with much more care than when solving for real or rational numbers. However, there are some basic things that we can do. The Division Algorithm (DA) is the procedure that best resembles general division. EDA yields a remainder r which is either zero or otherwise positive, but still stricly less than the divisor. This is the best we can hope for when working with integers. DA also implies that every integer is uniquely represented in a given base b. -- One of the most important and interesting number sequence is that of the prime numbers. Primes are the building blocks of the integers, as we will see soon. They are the "atoms" of the natural numbers. Many simply stated questions about prime numbers are still unanswered, despite tremendous efforts. An active area of research is to understand the distribution of the primes on the natural number line. The Prime Number Theorem states that the number of primes less than a given number x is approximately given by x/log(x), and that this approximation gets better and better the larger x is.
  3. Homework for Monday, Sept. 25. : The Euclidean Algorithm (EA) is an efficient algorithm to compute the greatest common divisor of two positive integers. This is one of the most important and nicest algorithms ever invented. By back-tracking the EA, we can actually solve equations of the form ax + by = c for any integers a,b,c. This again can be used recursively to solve such linear equations in three or more variables (instead of just two!) However, that will become increasingly messier as the number of variables is increased. -- The Fundamental Theorem of Arithmetic (FTA) states precisely why the prime numbers are the building blocks of the integers. That each positive integer is a unique product of primes is good for many divisibility problems. Provided that we have the factorization of given integer, then we can spot immediately whether or not one divides the other. However, finding the factorization of a given integer is computationally very hard. In fact, determining whether or not it is a prime is hard enough, let alone finding the complete factorization.

Exam Info