Math. 301, Sec. 001, Fall 2017
Homework and Announcements
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
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)
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.
- 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
- 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.
- 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.
- 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.)
- Homework 4.
- Selected solutions from HW -- 4.
Be sure to grasp what the binomial theorem states for multiplying
out expression of the form (x + y)n. A good
way to obtain the first few values is to draw the "Pascals Triangle".
- 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.
- 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.
- 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.
- Homework 7.
- Selected solutions from HW -- 7.
NB! I am aware that the solutions of the HW problems
are in the back of the book, but they are quite sketchy. I want
you to "fill in the blanks" so to speak or "connect the dots" and
obtain complete solutions. -- I also plan to discuss some problems
on the review sheet.
- 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.
- 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.
- 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.
- Homework for Monday, Nov. 20.
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.
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.