A Mathematical Detective: Studying Number Theory at PROMYS
By Louis Kang
Imagine: Opportunities and Resources for Academically Talented Youth
Center for Talented Youth, Johns Hopkins University, March/April 2003
When do mathematics, having fun with kids your age, and a beautiful college campus
unite in an unforgettable six-week experience? Why, during the Program in
Mathematics for Young Scientists, of course. Held at Boston University, PROMYS
(pronounced "promise") takes students on a journey through the dazzling depths of
various branches of mathematics, with a focus on number theory. To its lucky
participants, mathematics is never the same again.
The Whole Truth of Whole Numbers
I had a hint of the insights I would gain at PROMYS within a couple of hours of
arriving in Boston, when I heard a conversation between a counselor and a participant.
When the student asked about the purpose of studying number theory, the counselor
answered that number theory is mainly a method of teaching us scientific thought and
problem-solving. I had never thought of math that way.
The program unfolded over the next week into a daily routine. Every morning, there
was a lecture by Professor Glenn Stevens, who has an exceptional gift for explaining
complex ideas. Number theory, we learned, is concerned with whole numbers and
equations that can be solved using only whole numbers.
During class, we created a list of the most basic statements that completely describe
whole numbers. We eventually narrowed the list to eight items, and out of that list, a
whole division of mathematics unfurled. Our list included
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multiplicative identity: 1 times any number equals that number,
additive identity: 0 plus any number equals that number, and
additive inverse: –a + a = 0.
We assumed, without listing them, the properties of equality: a = a; if a = b, then b =
a; if a = b and b = c, then a = c; and if a = b, then a + c = b + c.
For homework, we were given problems to attempt and mathematical proofs to solve.
One unique characteristic of PROMYS is that we received problems on new topics a
few days before they were covered in class, allowing us to make explorations and
discoveries on our own.
The proofs on the problem sets required an approach to math that I had never
encountered. I was given a statement and had to show that it is true (or sometimes
false), using only the eight axioms on the list created in class, as well as a lot of
ingenuity and reasoning. It was like being a detective; I saw what the outcome was but
had to find the reason and the circumstances that led to it. I remember roaring in
indignation when I had to prove that a ∙ 0 = 0, where a is any integer; that –a = –1 ∙ a;
and that -(a ∙ b) = -a ∙ b. Once you have proven statements like these, you gain a new
level of understanding that is unreachable otherwise. Now I knew why Professor
Stevens wrote on the blackboard during the first lecture, "Simple ≠ Easy."
Unexpected Connections
One of my first experiences with number theory at PROMYS was with Diophantine
equations, which are like ordinary equations, but with an interesting, almost sadistic
twist. Suppose I were to tell you to find values x and y that satisfy the equation 29x +
11y = 1. That's not so bad. But suppose I require that x and y be integers. Now it gets
tricky. Fortunately, there are powerful algorithms and techniques to solve these
Diophantine equations and more complex ones like x2 - 22y2 = 1, y ≠ 0. They are
derived from various different topics in number theory that, on the surface, seem to
have no relationship to Diophantine equations.
When Dr. Andrew Wiles proved Fermat's Last Theorem, a famous number theory
problem, he embarked on a dizzying journey through four dimensions and along
elliptical curves to show that there are no solutions to the Diophantine equation x n +
yn = zn, n > 2. Studying number theory showed me that mathematics has exquisite
symmetry and consistency, and that different branches of it are strongly and
fundamentally interconnected.
As the days passed, our investigation of number theory expanded. For example, we
looked at prime numbers and searched for theorems of unique prime factorization. To
do this, an old gem had to be revisited and proved: the algorithm for long division that
we learned in 5th grade. Long division and prime numbers? Related? Surprisingly but
most definitely yes. To prove unique prime factorization, you first need to prove that
for any two whole numbers, you can divide one by the other and get a quotient and
remainder that are both integers. With this, you can then prove a statement about
solving Diophantine equations; next, you incorporate this proof into a theorem on
divisibility. Finally, you use this last theorem to prove unique prime factorization.
Phew! As you can see, many grand ideas in number theory are results of numerous
smaller concepts that lead up to it, and the glue that connects these steps together is
mathematical reasoning.
Then, we took the knowledge gained from studying whole numbers and integers, and
extended it to other number systems with similar properties. Thus, we used
generalization, an integral technique of mathematics, to find the most fundamental
way of stating ideas. For example, we found that unique prime factorization holds also
in Z[x], which is the system of all one-variable polynomials with integer coefficients,
and also in the Gaussian integers, a system which includes all numbers in the form a +
bi, where a and b are integers and i = √ – 1. Thus, we can divide polynomials and
Gaussian integers and factor them into primes much like we do with integers.
However, there are quirks here and there. Take a number system called Z[√ – 5] for
instance: it has neither a division algorithm nor unique prime factorization! It might
sound strange, but then, number theory is full of amazing and baffling details.
The World of PROMYS
In addition to lectures, there were labs where we worked in groups on problems of our
choice, mini-lectures given by counselors and special guests, and advanced classes for
students who return to PROMYS for a second year. Although first-years are allowed
to attend second-year classes, I was too busy with my classes to even sit in on one of
them, whereas other more advanced first-years eagerly took the opportunity.
However, I went to many mini-lectures and guest lectures on topics ranging from
combinatorics to graph theory, which I enjoyed tremendously.
Even with all of this math and work, we did not forget to enjoy our summer and have
fun. I started each day with a jog on a picturesque bike trail along the Charles River,
only a block away from the dorm. I also joined in various games of Ultimate Frisbeeincluding the annual game against another summer program called Research Science
Institute (RSI), which is held at MIT. We saw Fenway Park, took the subway to
Quincy Market, and went to Harvard Square. We visited MIT and Harvard, which are
located just across the river.
I still feel a tinge of wonder when I glance at number theory proofs in my PROMYS
notebook. Now, when I look at a math problem in a contest, I have a better
understanding of how I should attack it (a great help on timed competitions, where the
speed of your thinking greatly affects your outcome). I have found that many
problems include number theory, or at least can be solved more easily with a number
theory background. Studying it has definitely given me a new insight into and
appreciation for mathematics.
Louis Kang is a sophomore at Eastern Regional High School in New Jersey. He is
interested in both mathematics and biological sciences, and he has volunteered for the
past two summers in a neuroscience research laboratory at the University of
Pennsylvania.