# PRIMES: Shyam Narayanan's Story

My love for mathematics initially blossomed from competitions. From MATHCOUNTS to USAMO, such competitions blended the fun of math, the spirit of competition, and the friends who shared so many (academic and nonacademic) interests with me. Anytime I was competing, there was nothing else I would have wanted to do instead. But I never gave much consideration to mathematical research until I heard about the PRIMES program. When I was selected, I was excited but very surprised. I had never done any real scientific research, and even though I knew students who conducted scientific research in high school, I never thought those students would include me. I was, after all, a competition guy.

But I soon realized math Olympiads and mathematical research were much more similar than I had assumed. When I proved my first new theorem, I was shocked to realize that number theoretic techniques I learned through competition math easily carried over to the research world. Thanks to the PRIMES program, I was able to let my love for mathematical research blossom as well.

Research is challenging. It leaves you despairing after being unable to solve a conjecture for weeks. It leaves you confused as to where you can go next. It leaves you angry after you realized one tiny hole in your solution has caused your theorem to come crashing down. However, math has a nice property of branching out into many possibilities. While one method of solving a problem may not work, there are dozens of paths out there. With the help of my mentor, David Corwin, I was able to discover the numerous ways I could approach my problem, and while my methods were not the originally intended ones, they were successful nonetheless. For example, while I initially looked for empirical patterns among strong pseudoprimes for the Miller-Rabin Primality Test, I noticed one theorem I proved revealed an additional pattern I could not find from my empirical results.

While PRIMES is challenging and time-consuming, the effort required pales in comparison to what can be achieved. PRIMES teaches you the true beauty of mathematics and shows you new results are always possible: you just have to look in enough places. PRIMES also gives you the understanding and confidence required for scientific progress. PRIMES is the perfect opportunity for students to pursue mathematical research, whether they are national science fair champions or have never done research before.

**Shyam Narayanan **
worked on the project
Improving the Speed and Accuracy of the Miller-Rabin Primality Test under
the mentorship of David Corwin and the supervision by Dr. Stefan Wehmeier and Dr. Ben Hinkle. The project was sponsored by MathWorks.

**Email us:**
primes@math.mit.edu