September 29, 2006

Rose-Hulman Institute of Technology

5500 Wabash Avenue

Terre Haute, IN 47803

Professor Roger Lautzenheiser:

We would like to submit our revisions, following the recommendations of the referee, of the paper "*On Large Rational Solutions of Cubic Thue Equations: What Thue Did to Pell.*" We thank the referee for useful comments and criticisms. However, we decided to keep our exposition on elliptic logarithms so that its construction would not seem too mysterious to the uninitiated reader; we hope the length of our discussion will not be looked upon too unfavorably by the referee. We remind you that there are two graphics files, cubic.pdf and group_law.pdf, that need to be included in order for the paper to compile.

As you requested, here is an abstract for our paper:

In 1659, John Pell and Johann Rahn wrote a text which explained how to find all integer solutions to the quadratic equationAs you also requested, the biographical sketches of the authors read as follows:u. In 1909, Axel Thue showed that the cubic equation^{2}- d v^{2}= 1uhas 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^{3}- d v^{3}= 1(uwhich increase without bound as_{n}, v_{n})nincreases without bound. Such cubic equations are birationally equivalent to elliptic curves of the formy. The rational points on an elliptic curve form an abelian group, so a "large" rational point^{2}= x^{3}- D(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

u. We explain why^{N}- d v^{N}= 1N = 3is 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).

**Jarrod Anthony Cummingham** (jarrod2001@yahoo.com) 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** (nho@mills.edu) 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** (jonam@nsm.buffalo.edu) 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** (thomas_nikia@msn.com) 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.

We would like to remind you that the work for this paper was completed during the summer of 2004 during an REU at SUMSRI / Miami University under the guidance of advisor Dr. Edray Goins. For all future correspondence, the address of the corresponding author is:

Thank you for your time.

Jarrod Anthony Cunningham

Nancy Ho

Karen Lostritto

Jon Anthony Middleton

Nikia Tenille Thomas