In 1659, John Pell and Johann Rahn wrote a text which explained how to find all integer solutions to the quadratic equation u2 - d v2 = 1. In 1909, Axel Thue showed that the cubic equation u3 - d v3 = 1 has finitely many integer solutions, so it remains to examine their rational solutions. We explain how to find "large" rational solutions i.e., a sequence of rational points (un, vn) which increase without bound as n increases without bound. Such cubic equations are birationally equivalent to elliptic curves of the form y2 = x3 - D. The rational points on an elliptic curve form an abelian group, so a "large" rational point (u,v) maps to a rational point (x,y) of "approximate" order 3. Following an idea of Zagier, we explain how to compute such rational points using continued fractions of elliptic logarithms. We divide our discussion into two parts. The first concerns Pell's quadratic equation. We give an informal discussion of the history of the equation, illuminate the relation with the theory of groups, and review known results on properties of integer solutions through the use of continued fractions. The second concerns the more general equation uN - d vN = 1. We explain why N = 3 is the most interesting exponent, present the relation with elliptic curves, and investigate properties of rational solutions through the use of elliptic integrals. This project was completed at Miami University, in Oxford, OH as part of the Summer Undergraduate Mathematical Sciences Institute (SUMSRI).

Author Bio

Jarrod Anthony Cummingham graduated from the University of South Alabama with a B.A. in Mathematics and Statistics. He is currently working towards a degree in Aviation Technology, after which he plans to enroll at the Florida Institute of Technology to earn a Ph.D. in Space Science.

Nancy Ho graduated from Mills College with a B.A. in Mathematics and a minor in Computer Science. Currently, she is studying toward a M.A./Ph.D. in Mathematics at the University of Oklahoma.

Karen Lostritto (karen.lostritto@yale.edu) is a first year Ph.D. student at Yale University in Computational Biology and Bioinformatics. She graduated magna cum laude from Brown University with an A.B. in Mathematics.

Jon Anthony Middleton is a first-year Ph.D. student in Mathematics at the University of California, San Diego. He received a B.S. in Mathematics from SUNY Buffalo. His interests currently reside in algebraic topology and spectral geometry.

Nikia Tenille Thomas is a first year graduate student at Howard University pursuing a Ph.D. in (pure) Mathematics. She received her bachelor's degree in Mathematics from Morgan State University. Before entering graduate school, she took a year off to teach in the Baltimore public school system.