Type of Culminating Activity
Master of Science in Mathematics
Dr. Marion Scheepers
Pell's equation has intrigued mathematicians for centuries. First stated as Archimedes' Cattle Problem, Pell's equation, in its most general form, X2 – P • Y2 = 1, where P is any square free positive integer and solutions are pairs of integers, has seen many approaches but few general solutions. The eleventh century Indian mathematician Bhaskara solved X2 – 61 • Y2 = 1 and, in response to Fermat's challenge, Wallis and Brouncker gave solutions to X2 – 151 • Y2 = 1 and X2 –313 • Y2 = 1. Fermat claimed to posses a general solution, but it wasn't until 1759 that Leonard Euler published the first general solution to Pell's Equation. In fact, it was Euler who, mistakenly, first called the equation Pell's Equation after the 16th century mathematician John Pell. Pell had little to do with the problem and, though Pell made huge contributions to other fields of mathematics, his name is inexplicably linked to this equation.
One natural generalization of the problem is to allow for 1 to be any integer k. This yields the Pell-Like equation X2 – P • Y2 = k, where P is any prime and k is any integer. In fact, on his way to the solution of X2 – 61 • Y2 = 1, Bhaskara solved many Pell-Like equations; although at the time this was not his goal.
Neglecting any time considerations, it is possible, using current methods, to determine the solvablility of all Pell-Like equations. Whereas some have claimed that these methods solve the problem, we shall illustrate that a decision as to the solvability of many Pell-Like equations is computationally unfeasible.
From a computational standpoint, there are two fundamental questions associated with Pell-Like equations. First, is there an efficient means to decide if solutions exist? Second, if a particular Pell-Like equation is solvable, is there an efficient means to find all solutions? These are, respectively, the Pell-Like decision and search problems.
The problem of finding an efficient solution to the Pell-Like decision and search problems, for all Pell-Like equations, remains unsolved. In what lies ahead we hope to shed some illuminating light on these problems and give a partial solution. Our tools are Modular Arithmetic, Gauss' Quadratic Reciprocity Law, and the theory of Continued Fractions. We review these as well as historical efforts on these problems in Chapter 1.
Once we have developed the necessary theory for the Quadratic Reciprocity law and the theory of Continued Fractions, we will use these ideas in Chapter 2 to further develop a partial criterion for the solvability of Pell-Like equations. Using the Quadratic Reciprocity law we will develop a series of tests that efficiently decide the unsolvability of many Pell-Like equations. Using the Theory of Continued Fractions, we will develop in Chapter 3 the necessary tools to solve the Pell Like search problem for a specific subset of all Pell-Like equations. We show that all solutions taken on by convergents in the continued fraction expansion of √P are taken on within the first two iterations of the period.
Chapter 4 presents cryptographic applications of Pell-Like equations. We will define and prove the existence of a cryptographic group and then discuss its applications to many types of cryptosystems. We conclude with a discussion of a cryptographic attack that uses the Theory of Continued Fractions.
For all results from classical Number Theory which we do not prove we refer the reader to . For those results from the Theory of Continued Fractions that we do not prove, we refer the reader to , , and .
Smith, Jason, "Solvability Characterizations of Pell Like Equations" (2009). Boise State University Theses and Dissertations. Paper 55.