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Pell’s Equation

Fermat claimed to posses a general solution, but it wasn't until 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.


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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. 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.

Solving the Pell Equation

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.

Number Theory: Diophantine Equation: ax+by=gcd(a,b)

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. 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.

Pell equation - Encyclopedia of Mathematics

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.


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  • 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 [2]. For those results from the Theory of Continued Fractions that we do not prove, we refer the reader to [8], [2], and [9].

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    Skip to main content. Degree Title Master of Science in Mathematics. Abstract Pell's equation has intrigued mathematicians for centuries.


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