Mathematicians like the mathematics that most people are not aware about.
The course answers to the question What is the mathematics that
mathematicians like?
Students who will also find this mathematics attractive will be helped to
enter it.
Each lecture is aimed to surprise and
reshape the students' perception of mathematics.
The course is organized as a collection of short, important, and self-contained
subjects, which are mostly independent from each other. They are picked up from
various parts of mathematics: number theory, geometry, set theory,
combinatorics, topology, optimization theory, etc.
and complement the content of advanced mathematical courses.
No preliminary knowledge of advanced mathematics is required.
The course is targeted towards students of the following three types:
- freshmen who have not yet decided what major to choose
- beginning math majors
- junior and senior math majors
A freshman would be helped to decide whether mathematics deserves her/his
attention and efforts,
whether to become a mathematics major, or to choose another major subject.
A beginner math major would learn plenty of nice mathematics and
would be helped to decide if she/he is ready to join the
honors program in mathematics immediately.
The course will provide students with a rich collection of classical
mathematics of prime importance, which otherwise may not reach students,
because is not included in the main math courses.
The Enjoyment of Mathematics by Hans Rademacher, Otto Toeplitz.
The book is available from Amazon
for about $ 13.
Certainly, the course will not follow the book section by
section, but the style would be similar.
From the introduction to this book:
We select a sequence of subjects, each one complete in itself,
none requiring more than an hour to read and understand.
The subjects are independent, so that one need not remember
what has gone before when reading any chapter.''
The table of contents:
- The Sequence of Prime Numbers
- Traversing Nets of Curves
- Some Maximum Problems
- Incommensurable Segments and Irrational Numbers
- A Minimum Property of the Pedal Triangle
- A Second Proof of the Same Minimum Property
- The Theory of Sets
- Some Combinatorial Problems
- On Waring's Problem
- On Closed Self-Intersecting Curves
- Is the Factorization of a Number into Prime Factors Unique?
- The Four-Color Problem
- The Regular Polyhedrons
- Pythagorean Numbers and Fermat's Theorem
- The Theorem of the Arithmetic and Geometric Means
- The Spanning Circle of a Finite Set of Points
- Approximating Irrational Numbers by Means of Rational Numbers
- Producing Rectilinear Motion by Means of Linkages
- Perfect Numbers
- Euler's Proof of the Infinitude of the Prime Numbers
- Fundamental Principles of Maximum Problems
- The Figure of Greatest Area with a Given Perimeter
- Periodic Decimal Fractions
- A Characteristic Property of the Circle
- Curves of Constant Breadth
- The Indispensability of the Compass for the Constructions
of Elementary Geometry
- A Property of the Number 30
- An Improved Inequality
Notes and Remarks
This is the only textbook which is recommended.
The whole book will not be covered, but this is an excellent
source for reading.
- Score of level 4 or better on the math placement exam,
- or MAT 123 passed with a grade C or better,
- or a permission by the instructor.
Priority given to students in the University's honors programs.
Notice that the prerequisites do not include any Calculus. High-school
algebra suffices.
Though the course is not easy: as any mathematics, it relies on proofs.
This is why MAT 200: Logic, Language and Proof is
an advisory co-requisite.
Oleg Viro
Professor,
Ph.D. 1974, Doctor Phys-Mat.Sci. 1983, both from Leningrad State
University
Arrived at Stony Brook in 2007.
Office: Math Tower 5-110
Phone: (631) 632-8286
Email: oleg.viro AT math.stonybrook.edu
Web page: www.math.stonybrook.edu/~oleg
Research fields: Topology and Geometry,
especially low-dimensional topology
and real algebraic geometry.
Mondays and Wednesdays 5:30pm - 6:50pm in Library W4530,
Mondays and Wednesdays 7:00pm - 8:00pm in Math Tower 5-110.
The final grade will be based on
- Homeworks 20%,
- Midterm 40%,
- Final exam 40%.
Homeworks will be assigned once a week. A typical homework will consist
of two - three problems.
In MAT 150, the main purpose of homeworks is to keep
students aware about the material of lectures: definitions, examples,
statements of theorems, some proofs. More creative challenging problems
(e.g., compose a problem or generalize a theory given in a lecture)
will be proposed, but rarely and only as non-compulsory bonus problems.
The exams, both midterm and final will serve the same purpose: to verify
whether the material is familiar.
If you have a physical, psychological, medical, or learning disability that
may impact your course work, please contact
Disability Support
Services or call (631) 632-6748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential.
Students who require assistance during emergency evacuation are encouraged
to discuss their needs with their professors and Disability Support
Services. For procedures and information go to the
Evacuation
Guide for People with Physical Disabilities.
Each student must pursue his or her academic goals honestly and be
personally accountable for all submitted work. Representing another
person's work as your own is always wrong. Faculty are required to report
any suspected instances of academic dishonesty to the Academic Judiciary.
For more comprehensive information on academic integrity, including
categories of academic dishonesty, please refer to the
academic judiciary
website.
Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn.
The sequence of prime numbers
The first lecture followed closely the first chapter of the textbook.
After the lecture its content was
presented
in a more formal style. In the Homework 1, a comparison of the two texts
was requested. More specifically, the proofs of theorems 1 and 3 had to be
done.
The statement and proof of theorem which was
called Theorem 3 in
Lecture 1 were not formally distinguished
in the textbook. The proof was not quite clear. The homework required to
find them in the textbook, and to answer to the following two questions:
What is not said in the proof? What is number $p$ in the proof of Theorem 3?
and list the differences between the proofs in the lecture and textbook.
The number $p$ is not defined in the textbook. One can guess what $p$ is.
Moreover, the numbers in the formula that defines $M$ are not specified
either. They cannot be the same as in the proof of Theorem 3 in
Lecture 1.
The textbook is a beautiful classical book. A good mathematical book
may, and even should, be somewhere unclear. This gives to a student an
opportunity to think, fix the vague piece and improve self-esteem.
The third problem (bonus problem) in Homework 1:
What sets of primes can you prove to be infinite?
-
the sets of primes in arithmetic series $4,8,12,16, \dots,4n,\dots$;
$1,5,9,\dots,4n+1,\dots$; $6,10,14,18, \dots,4n+2\dots$;
$3,8,11,\dots,4n-1,\dots$?
-
the sets of primes with the last digit $3$ or $7$ (i.e., having the last
digit $3$ or $7$ in the decimal notation)?
Pythagorean triples
The lecture used a new technique: a pdf computer
presentation was used.
This is a screen presentation file. It consists of 80 pages.
Pages next to each other differ usually just by one sentence.
Read it on a computer screen. Switch to pdf-viewer to page-wise
regime.
The pdf file was used as a background for xournal. This allowed
to write over the screen. Almost all the proofs were inserted
during the lecture.
Now if any of the numbers $2, 5, 8, 11,\dots$ is split into its prime
factors, at least one of the prime factors must again be one of $2, 5,
8, 11,\dots$. For example, $14 = 2\cdot7$ has the factor $2$ and
$35 = 5\cdot 7$
has $5$. To show that this is always true, we note that each of the
prime factors must be either a multiple of $3$, in the sequence $1, 4,
7, 10,\dots ,$ or in the sequence $2, 5, 8, \dots$. The only multiple of
$3$ that is a prime is $3$ itself, and it does not divide our chosen number.
If all the prime factors were of the kind $1, 4, 7, 10, \dots$, then by
the above remark (1) our number would again be of this kind.
Since it is not of this kind, at least one of its prime factors must be
one of the following $2, 5, 8, 11,\dots$.
We can now proceed as in Euclid's proof except that we consider
$$2\cdot3\cdot5\cdot7\cdot11\dots p - 1 = M$$
in place of
$$2\cdot3\cdot5\cdot7\cdot11\dots p + 1 = N$$
$M$ is a multiple of $3$ decreased by $1$, which means that it is one of
$2, 5, 8, 11, \dots$. Just as with $N$, it is clear that $M$ is not
divisible
by any of the primes $2, 3, 5, 7, 11,\dots, p$. Either $M$ is a prime
(greater than $p$) or all its prime factors are greater than $p$. In
the latter case at least one of the prime factors is one of $2, 5,
8,\dots,$
and hence in both cases we have found a prime of this kind that
is greater than $p$. Therefore the sequence $2, 5, 8,\dots$ contains
an infinite number of primes.