Zubin Mukerjee, at right, with other members of the Upstate New York All-Star Math Team at the national high school math tournament, ARML, in June |

Zubin Mukerjee, a veteran leader of Albany Area Math Circle who has also organized and led a satellite math circle of his own for younger students, has blazed yet another a new trail for others to follow.

Zubin, a Guilderland High School senior, who is taking advanced classes in math and economics at SUNY Albany, and his co-author, Uthsav Chitra from Delaware, have won semifinalist honors in a prestigious national research contest for high school students. Zubin and Uthsav worked on an original research project in number theory, "Random Involutions and the Number of Prime Factors of an Integer,"under the guidance of a mentor at PROMYS in Boston last summer.

Research presents new challenges as well as new rewards compared to the contest problems with which many math circle students are familiar. When you work on a contest problem, it may be very hard, but you KNOW that you are working on a problem that somebody else has already solved and that there must be a clever elegant solution to the problem. It is indeed exciting to have the Aha! moment when you find the solution to a contest problem, but such moments pale compared to those you can experience in math research, the thrill of discovering an answer to a problem nobody else has ever found before. Sometimes the results are negatives ones, not exactly the ones you were hoping for originally, but even those disappointing results can provide important clues to promising new lines for exploration.

It takes passion, perseverance, and luck to find original new research results, far moreso than in contests. When doing original research, there are no guarantees at the outset that the problem will even have a solution at all, let alone that it will yield interesting results worth sharing with others. Even once the problem is solved, it takes excellent writing skills to write up your research results in a way that will allow others to appreciate the importance and validity of what you have discovered. Zubin's years of helping to write power round solutions for our math circle teams as well as his prize-winning entries in history day competitions have certainly polished his expository writing skills.

Here are the abstract and executive summary for Zubin and Uthsav's research project.

**Abstract:**

*For hundreds of years, mathematicians have tried to find good approximations for the function d(n), which counts the number of prime factors of an integer n. In this paper, we examine*

*using random involutions to approximate d(n) by comparing the number of fixed points of a*

*random involution on F*

_{2}^{2g(n)}to the number of fixed points of a specific involution, τ(n). We find and prove that the expected number of fixed points of a random involution converges, so that d(n) cannot be approximated using this method; moreover, we use this to show that the involution τ(n) is not random, as it has more fixed points than a random involution.**Executive summary:**

*The natural numbers are perhaps the most familiar to humans. They are the counting numbers: 1,*

*2, 3, etc. A divisor of a number is something that divides evenly into that number. For example, 3*

*and 14 are divisors of 42, but 42 is not a divisor of 3 or 14. A prime number is a natural number*

*whose only positive divisors are 1 and itself. The first few primes are 2, 3, 5, 7 ... there are also*

*infinitely many of these. There is a well-known function that returns the number of prime divisors*

*of a number n, given that number. We denote this function d(n). Our goal in this project is to*

*further research on modeling d(n).*

*Our mentor proposed a possible method of modeling d(n) by looking at special functions called*

*involutions that act on the surface of modular curves. In particular, we studied the involution*

*τ*

*(n), which is related to d(n), by comparing it to random involutions. We were able to conclude,*

*through a series of proofs and derivations as well as some graphical analysis using Mathematica,*

*that d(n) cannot be modeled by*τ

*(n) and that, as a result,*τ

*(n) is not random. In other words,*

*there is something special about*τ

*(n) that makes it so we can’t model d(n).*

*The consequences of this result are not yet fully clear. Nevertheless, this result can lead the way to*

*studying other types of involutions, some of which may be able to model d(n). An accurate model*

*for d(n) would be incredible, as it would make finding the prime factorization of large numbers*

*much easier; this would have many applications in cryptography and computer science. Much*

*research remains to be done on involutions though; perhaps one day, a closed-form expression for*

*d(n) will be found through random involutions.*

Zubin and Uthsav's research mentor was Dr. Kirsten Wickelgren, an American Institute of Mathematics fellow at Harvard University. Here is a link to a copy of the background document including the problems she suggested they investigate as well as definitions of some key concepts and a helpful list of the supplementary references with which she initially launched them on their way. If you are interested in understanding more about their work, you may want to take a look at those references yourself. Students who have not yet studied much number theory will also find the

*Art of Problem Solving'*s textbook on introductory number theory very helpful. [Added later: Zubin also passed along another recommendation of a classic number theory book, Hardy & Wright's

*Introduction to the Theory of Numbers,*endorsed by PROMYS Director Glenn Stevens as "clear and concise." Zubin also notes that Hardy & Wright cover many topics in number theory in their book, some relevant to their project and some not.]

You will also note that Zubin and Uthsav used Wolfram

*Mathematica*computer software to help create graphs to give them insights into their problem analysis. Thanks to Wolfram's sponsorship of contests such as American Regions Math League (ARML), Harvard-MIT Math Tournament (HMMT), and Princeton University Math Contest (PUMaC), Zubin and all our veteran math circle students who have participated in one or more of those contests have received free student licenses to use this very powerful software. Those licenses will remain valid as long as they are students, including college and graduate school years ahead.