Number Theory

Math. 301, Sec. 001, Fall 2017

Homework and Announcements

Urgent stuff

Plan for classes until Thanksgiving My colleague, Prof. Neil Epstein, will lecture on my behalf on Wednesday Nov. 15th on my behalf and hand back your homework. (ii) The class on Monday Nov. 20th is tentatively cancelled (if need be, we will make it up later.) -- Have a nice Thanksgiving.!

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 WEBSITE 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.
  4. Homework for Monday, Oct. 2. : What we have discussed so far are some special forms of large prime numbers, the Mersenne primes and the Fermat primes. Note that neither is always prime! The Fermat primes are interesting for the mere fact that any two of them are always relatively prime. This gives us another way to prove that there are infinitely many primes. Namely, the sequence of Fermat primes is infinite, and they must therefore generate an infinite number of primes. -- Congruences do not present anything new per say, but the notation (due to Gauss) is very nice, since it shows that we can use congruences almost like an equality. That a is congruent to b modulo m means precisely that a and b are equal up to a multiple of m. -- (A good place to look at the properties of binomial coefficients is in Appendix B of the book. Also briefly look at the property of the Binomial Theorem in case you have not seen it before.)
  5. Homework for Tuesday, Oct. 10. : To solve linear congruences is exactly the same as solving ax + my = c in integers, except for the fact that we are only interested in the solution in x since any multiple of m we are simply not relevant. Hence, we know exactly how to solve a single linear congruence equation. The Chinese Remainder Theorem (CRT) tells us exactly when we can solve a system of linear congruences involving a single variable x, when all the moduli are pairwise relatively prime. When the moduli are general, then the system needs to be tweaked a little in order to apply the CRT. We also discussed how to solve nonlinear congruences and we started discussing some special interesting congruence relations, the first one due to Wilson, which give us an arithmetic (albeit not efficient) test of whether a number is a prime or not.
  6. Homework for Monday, Oct. 16. : So far we have discussed three interesting congruence relations, Wilson's Theorem, Fermat's Little Theorem (FLT) and Euler's Theorem. Wilson's Theorem is interesting since it yields a congruence formula that characterizes a prime number p. Although FLT holds for primes p it is by no means a characterization, since there are numbers that are not prime that yield similar congruence relations, the pseudo primes and absolute pseudo primes. One great thing with FLT is that it gives us a formula for solving linear congruence equations modulo primes. Before we were able to solve each case by using EA in reverse, but we had no formula to express the solution in general. Euler's Formula generalizes FLT to numbers that not necessarily prime, and it also give us a way to solve a linear congruence equation using a direct formula.
  7. Homework for Monday, Oct. 23. : Arithmetic or number theoretic functions are just functions from the set of natural numbers into itself. Multiplicative functions form an important class of arithmetic functions, since many algebraic properties of numbers can be given by means of multiplicative functions (e.g. a number is square iff the number of its divisors is odd). The summatory function of a multiplicative function is again multiplicative. Hence, both the tau and sum-tau functions are multiplicaive. The Euler phi function also turns out to be multiplicative. This yields a nice formula for the Euler phi function, which, in addition, tells us exactly the fraction of the numbers from one up to the given integer, are relatively prime to the integer.
  8. Homework for Monday, Oct. 30. : This week we did a little review, discussed some problems on the review sheet and then had a midterm on Wednesday.
  9. Homework for Monday, Nov. 6. : If we know the summatory function of an arithmetic function then (1) we know the arithmetic function completely, and we have a nice formula to get it back from the summatory function by using the Möbius function. The Möbius function is an important multiplicative function. (2) We also know that if the summatory function is multiplicative, then so is the given arithmetic function as well. This we obtain from the Möbius Inversion Formula that tells us how to obtain the arithmetic function back from its summatory function (not unlike the "Delta Dirac function" tells us how to get the function back from its integral.) Hence, we have that an arithmetic function is multiplicative if, and only if, its summatory function is multiplicaive. -- Primitive roots modulo m are those roots that will generate all other roots of the exponential equation. We know both from Fermat's Theorem and its generalized form of Euler's Theorem that there is always a solution and that guarantees that every integer relatively prime to m has an order modulo m. However, not every m has a primitive root although every integer relatively prime to m has an order modulo m. We did see, however, that if an integer n has a primitive root, then it has many of them, in fact φ(φ(n)) of them. But, we still have not discussed which integers have at least one (and hence many) primitive roots.
  10. Homework for Monday, Nov. 13. : The two main things we established were (1) If m has a primitive root, then we know exactly how many of them there are. (2) Every prime number has a primitive root. -- We also just started discussing quadratic residue modulo a prime number. We showed that the next logical step of congruence equations was to consider "square roots" of an integer modulo a prime number. An integer is called a quadratic residue modulo the prime if the integer has a square root modulo that prime, otherwise it is called a quadratic nonresidue modulo that prime. It turns out that exactly half of the numbers from 1,2,...,p-1 are quadratic resudue modulo p.
  11. Homework for Monday, Nov. 20. Euler's Criterion is our first theorem that describes when exactly a number is a quadratic residue mod p. We have discussed the Legendre symbol for primes and obtaind some criteria on when it is 1 and when it is -1. The outcome tells us precisely when the corresponding quadratic concruence equation has a solution and when not. The Lemma of Gauss is nice for many reasons, in particular it can be used to determine whether or not 2 is a quadratic residue modulo any odd prime.

Exam Info

Here below is a review sheet for the midterm to be held on Wednesday, October 25. -- Please note that this sheet contains questions that I could potentially ask on the midterm, but it is not a recipe for the midterm! The midterm will consist of 5 - 8 problems each involving some computation or a short proofs of a statements. Basically the midterm will be on stuff that we have covered up to and including last week, but not what we will cover this week.

Below are some drafts of solutions for some of the problems on the review sheet for the midterm. Please note that these are meant to present you the main ideas, but are not all fully written detailed solutions! -- Do attempt the problems here above first, before you read and study the solutions.

Below are handwritten solutions to the midterm exam, as I wanted to see them. The raw median score was 50/100. Note that I record only the number grade and not the curve I gave in class. -- Please, read over these solutions and compare to what you did yourselves.