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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

• multiplicative identity: 1 times any number equals that number,
• additive identity: 0 plus any number equals that number, and
• additive inverse: –a + a = 0.

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 xⁿ + 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 Frisbee-including 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.