Professional Preparation

Research Areas

Research Interests

1. Enhancing students interests and skills in mathematics.
I am the Director and Founder of AwesomeMath, which includes a summer program for students training to succeed at the Olympiad level, a correspondence-based lecture series for students continuing education, called AwesomeMath Year-round, and Mathematical Reflections, a free online journal focused primarily on mathematical problem solving. The main purpose of this initiative is to give students an opportunity to engage in meaningful learning activities and explore in detail areas in advanced mathematics. AwesomeMath's primary focus is on problem-solving. I use it as a tool to enhance students' interest and skills in mathematics.

I believe that there are two major parts in significant mathematics teaching and learning: higher concepts (introducing and developing new topics) and applying those concepts creatively to concrete problems (bringing life to the new topics). These two areas rely on each other, but we focus primarily on the latter. I feel that certain advanced mathematics topics are best introduced to young students by motivating the concepts through problems that encourage investigation. 

I am actively involved with mathematics competitions at the secondary and undergraduate levels.  I write and contribute questions for the American Mathematics Competitions examinations as well as the International Mathematics Olympiad and the W. L. Putnam competition. 

I would like to involve undergraduate students in my research.  My area of research is an ideal entrance point to research for talented students, since it does not require a lot of background while offering abundance of open problems.  In addition to the problems I can suggest, the students can also easily make their own conjectures, experiment by using Computer Algebra Systems, attack special cases, generalize and transfer ideas from one case to another, and learn and use various techniques while trying to resolve the problems.  Thus they would experience first hand all the phases and subtleties of doing original research.

2. Diophantine Analysis, with emphasis on Quadratic Diophantine Equations.
 This research area focuses especially on the study of the general Pell's equation, which is connected to problems from various domains of mathematics and science, such as Thue's Theorem, Hilbert's Tenth Problem, Euler's Concordant Forms, Einstein's Homogeneous Manifolds, Hecke Groups, and so forth.  I have obtained numerous original results such as:  

1. In the cases when the equation is solvable, I found an elegant explicit form for the solutions.  I then extended a result of D. T. Walker [The American Mathematical Monthly, vol. 74, no. 5, 1966, 504-513].
2. I proved partial results about the equation  and formulated conjectures regarding its solvability.
3. I obtained results about the equation , including those regarding the LMM algorithm of finding the fundamental solutions to the general Pell's equation based on continued fractions.
4. I devised two original methods of solving the equation .
5. I proved that numerous important quadratic equations have infinitely many solutions in integers.

I devoted special attention to the Diophantine representability of several interesting sequences of positive integers.  I introduced the concept of r-Diophantine representability of a sequence of positive integers and studied in an original manner the equations  by employing the special Pell's equation .  I have extensively studied the Diophantine representability of the Fibonacci, Lucas, and Pell sequences using methods of investigation different from and simpler than the ones already found in the literature.  I also studied the problem of Diophantine representability of generalized Lucas sequences, which I introduced, finding conditions under which their general solution is a linear combination with rational coefficients of the classical Fibonacci and Lucas sequences. 

I found numerous applications of the results above.  For example, I determined conditions under which the numbers  and  are simultaneously perfect squares for infinitely many positive integers n.  I discovered special properties of triangular numbers, such as proving that any positive rational number r, where is irrational, can be written as the ratio of two triangular numbers.  I extended some results pertaining to triangular numbers to polygonal numbers.

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