Scores on the Final: 180s(9), 170s(7), 160s(5), 150s(3), 140s(4), 130s(2), 120s(3), <100(3)
Grades in the Course: A(8), A-(4), B+(5), B(5), B-(5), C+(5), C-(1), D(2), F(1).
Visitation in my office Tuesday, May 18, Noon-12:30 or by appointment. I should be around "most" of the summer.
I hope you have an enjoyable and productive summer. -- BR
F 1/23 -- Second day of class. Distribution of Problem-solving handout
. A quick review of 1.1 and 1.2 in the book, including
the error in the definition of "algebraic number" on p. 7. We begin to
talk about divisibility. One error on my part, noted after class. When
I wrote 2/3 = 4/6 = 2004/3006, etc, apparently, I wrote (-2)/3, not
(-2)/(-3), which is what I intended. Since the class is about 1/3
math education majors, I will be brave enough to link "Chalking It Up"
. This is a guide for new mathematics teaching assistants that has
been used in more than 50 schools at one time or another over the last
20 years. Feel free to point out whenever I violate one of my own
suggestions for teaching!
M 1/26 -- The TA, Mr. Maosheng Xiong, was in class at the beginning, and
announced office hours on Wednesdays from 4-5 in 341 Illini Hall, or
by appointment. His email address is xiong@math.uiuc.edu. We finished
1.4 and began 3.1, talking about division and about the greatest
common divisors and prime numbers. Much of the material at the end of
section 3.1 is "cultural". That is, it is part of the course and
important to know, and you won't be tested on it. Ask me at any time
if you're unsure. (Anything we actually prove in class is likely to be
fair game for an exam.)
It's a boy!
Nathan Thomas Jacobs
Born to Angela Jacobs
Sure: By the definition, a, a^2 and a^3 are in arithmetic progression
(mod m) if a^2 -a = a^3 - a^2 (mod m) or a^3 - 2a^2 + a = 0 (mod m).
W 1/28 -- The first assignment, Homework
One, was distributed, and we completed sections 3.1 and 3.2.
Notation not in the book:
I write gcd(a,b) instead of (a,b), to denote
the greatest common divisor, in order to avoid confusion with the
point (a,b) in Cartesian coordinates.
For an integer a, aZ = {an : n in Z} is the
set of multiples of a, and aZ + bZ = {am + bn : m, n in Z} is the set
of integer linear combinations of a and b, so that Theorems 3.8 and
3.9 can be summed up as: aZ + bZ = gZ, where g = gcd(a,b).
I will always try to put additional notion on the webpage.
F 1/30 -- We finished 3.3, doing a few numerical examples of gcd,
and started a bit on 3.4. The textbook can be used for number theory
courses given in computer science departments. In general, I will skip
questions about the efficiency of algorithms. However, in this class
I gave a quick proof that if the ri's are given as in the
Euclidean algorithm on p.87, then ri > 2
ri+2. This means that the number of steps in computing
gcd(a,b) is bounded above by a linear multiple of the number of digits
of a. As computer algorithms go, this is a small number. Extra
notation to be
used: if n is a natural number and p is a prime, then vp(n)
is the power of p that divides n. For example, 24 = 233, so
v2(24) = 3, v3(24) =1 and vp(24) = 0
for every prime p > 3.
M 2/2 -- Printed version of webpage on large primes - see The Prime Pages .
The Fundamental Theorem of Algebra. Notation: if n is a
natural number and p is a prime, then vp(n)
is the power of p that divides n. For example, 24 = 233, so
v2(24) = 3, v3(24) =1 and vp(24) = 0
for every prime p > 3. Divisibility in terms of
vp(n). There will be a handout on Wednesday.
Homework One questions from my correspondents, and my answers
Q1. On number nine, does the phrase, "product of possibly a square
and a square free integer" mean show that you can always write it as a
product of two numbers, one will always be square free and the other may
or may not be a square?
W 2/4 -- Homework One due and handwritten solutions
distributed; Homework
Two passed out. Also, handwritten notes on vp(n),
including the formula for vp(n!), and the phone/email list.
Proved separately: if d = gcd(a,b), then a/d and b/d are relatively prime.
A1. I interpret it as follows: Either a number is squarefree, or it is
the product of a square and a squarefree integer. Of course, 1 is a
square, so there really isn't a meaningful alternative there.
Q2. On #10, do you want ALL possible values for gcd(b,c)? Also, are we
supposed to give one example of values for a,b,c,and n or do you want
us to try and figure out a formula for a,b,c,and n?
A2. Yes, all possible values of gcd(b,c), and for each particular value n
= gcd(b,c), find a single instance of (a,b,c) for which this holds,
not all instances.
F 2/6 -- Homework one not graded. In class, a discussion of the
factorization of 2^n - 1 and Mersenne primes and 2^n+1 and Fermat
primes. We move to 3.6 and solutions to linear diophantine equations.
Think about buying broccoli on a stick at $1.15 from an exact change
vending machine, when you only have dimes or quarters. I guess you had
to be there. Almost finished 3.6.
M 2/9 -- Homework 1 was returned, with supplemental comments. We went
through 3.6, problems 17 and 18, and started on 4.1.
W 2/11 -- Homework 2 due and handwritten solutions
distributed; Homework
Three passed out. We finished 4.1 amidst lots of numerical examples.
F 2/13 -- Your aching prof managed to make it through 4.2 with a hint
at 4.3. (Nothing broken, just a few bruises and strains. That will
teach me to pay attention where I walk.)
Sorry for the delay and brevity in writing up the last week's
classes. I'll put in more detail later if anyone asks.
M 2/16 -- A general remark: every math,
science or engineering major winds up being a teacher either formally
(e.g. high school, graduate TA, prof) or informally (e.g. explaining
some new software to your group at work.) I have written a TA training
guide used here and at many other schools at one time or another
(well, it's free), and you can find it here:
"Chalking It Up".
You'll be able to judge for yourself how well I meet my own
guidelines. We started the Chinese Remainder Theorem. And the Informal
Early feedback forms were distributed.
W 2/18 -- Homework 3 due and handwritten solutions
distributed; Homework
Four passed out. About half the class period involved working
through various problems from the third homework. We started on 4.4.
F 2/20 -- A handout on the theme of 4.4 if not the details: lifting
solutions of congrunces mod pk to congrunces mod
pk+1. The full result is deferred until later. The short
version is that for equations such as xn is congruent to a,
this lifting can always be done if n is not a multiple of p. If n is a
multiple of p, strange things happen. We began 6.1, with a proof of
Wilson's Theorem, and an indication of why Fermat's Little Theorem
should be true.
M 2/23 -- More on Wilson's Theorem (a bit) and Fermat's Little Theorem
(a lot). Plenty of numerical examples. A small discussion of
pseudoprimes. Speed and handwriting adjusted, as per suggestions from
the informal course reviews.
W 2/25 -- Homework 4 due and handwritten solutions
distributed; Homework
Five passed out. A short discussion of the examples and an
introduction to the Euler phi function. We also started using the
terminology ordm(a) to denote the smallest positive integer
k so that a^k is congruent to 1 (mod m). This is in the book, but later.
F 2/27 -- More on Euler's Theorem and the behavior of the Euler phi
function of mn and ordmn(a), when m and n are relatively
prime. Lots more numerical examples. Everything is beginning to tie
together nicely.
M 3/1 -- A completion of the proof that the sum of phi(d), when taken
over all divisors d of an integer n, is just n.
More on the Euler phi function and, more generally, on
multiplicative and completely multiplicative functions.
W 3/3 -- Homework 5 collected, and solutions distributed. A brief
introduction to the sigma and tau functions, which count the sum of
the divisors of n and the number of divisors of n, respectively.
These are also multiplicative functions. (This is in section 7.2.)
The discussion of these two days will not be on the first test.
F 3/5 -- Review discussion for the first test, homework 5 returned,
together with supplemental handout on the homework.
M 3/8 -- Test 1, in class.
W 3/10 -- Test 1 returned, with some discussion. More on arithmetic
and multiplicative functions. Homework
Six distributed.
F 3/12 -- More on multiplicative functions and Dirichlet products,
with a handout to be given on 3/15. A proof that if q is a factor
of 2p -1, where p is prime, then q = 2kp+1 for some k.
This is in the book.
M 3/15 -- A four page handout on multiplicative functions with a little
more on Chapter 7. A start on Chapter 9, primitive roots, with lots of
numerical examples.
W 3/17 -- Homework 6 solutions passed out and Homework
Seven distributed. Getting very close to the proof of the
existence of primitive roots.
F 3/19 -- A smallish class, so no new material. Homework 6 returned
and discussed to a bit, and some hints as to the rest of the semester.
Spring Break
Birth Announcement
on March 11th, 10:36pm
8 lbs 12 oz 21 inches
M 3/29 -- Homework 6 returned to those who were not there on
the 3/19. Supplemental solutions distributed, plus a three
page update on the search for large primes. A handout on primitive
roots mod 11.
We also proved the existence of primitive roots for odd primes.
This week was all pretty heavy-duty.
W 3/31 -- Homework 7 solutions passed out and Homework
Eight distributed. Some side discussions and implications for
the existence of primitive roots, and the non-existence, except
possibly for powers of primes or twice an odd prime power.
F 4/2 -- A handout on primitive roots mod 25.
The last day of primitive roots, finishing the proofs in
section 9.3. We shall move on the quadratic residues forthwith.
A student writes:
I am completely lost with numbers 9 and 10 in homework 8. Could you
give me a hint on how to start those two problems?
M 4/5 -- Introduction to quadratic reciprocity -- what are the
squares (mod p). Basic properties of the Legendre symbol.
W 4/7 -- Homework 8 solutions passed out and Homework
Nine distributed. More of Chapter 10 discussed, up through
Gauss' Lemma.
F 4/9 -- We finish all preparations for the proof of Quadratic
Reciprocity, on Monday 4/12. Homework 8 returned, together with
additional comments.
M 4/12 -- Quadratic reciprocity is proved! You are now in the
direct lineage of Gauss.
W 4/14 -- Homework 9 solutions passed out and Homework
Ten distributed. We begin our discussion of Diophantine
Equations, both in the methodology of the book, and via the
"point slope method". This is the last homework on which Test 2
will be based.
Apologies for a garbled hypothesis on HW 10 #9: the condition should
be that the Legendre symbol (a/p) = -1, not gcd(a,p) = -1. In
compensation, this new problem is now extra credit, everyone gets
1 point for #9 in any case.
F 4/16 -- Handout on the "point slope method" and examples. Plus
Fermat's Proof by Infinite descent of the nonexistence of solutions to
x4 + y4 = z2.
M 4/19 -- Homework 9 is returned, with supplemental discussion handout and
examples. We begin to look at the Diophantine equation x2 +
2y2 = ± 1, and its relation to approximation of
irrationals, continued fractions and certain sequences of integers.
W 4/21 -- Homework 10 is collected, and solutions are distributed.
There is a 4 page handout "Fun with the equation x2 +
2y2 = ± 1", which covers the discussion of 4/19 and 4/21.
F 4/23 -- Homework 10 is returned (only 20/36 papers, so I could
grade it quickly!), and supplemental discussion is distributed.
The main mathematical theme, after going through some review,
is the characterization of sums of two squares of integers (Theorems 13.3
through 13.6.) The material covered this week will not be on
the second exam.
M 4/26 -- Review for the second exam.
W 4/28 -- Second Hour Exam.
F 4/30 -- Second Hour exam returned and discussed. New material
(not per se on the Final): A proof of Hensel's Lemma (Th. 4. 14.)
M 5/3 -- Encryption and its connection to number theory.
The ICES forms were distributed.
W 5/5 -- Summing up for the semester and a few, last, neat things in
number theory.
F 5/14 -- Final Examination, 8-11 am.