This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: THE PELL EQUATION 1. Introduction Let d be a nonzero integer. We wish to find all integer solutions ( x,y ) to (1) x 2 − dy 2 = 1 . 1.1. History. Leonhard Euler called (1) Pell’s Equation after the English mathematician John Pell (16111685). This terminology has persisted to the present day, despite the fact that it is well known to be mistaken: Pell’s only contribution to the subject was the publication of some partial results of Wallis and Brouncker. In fact the correct names are the usual ones: the problem of solving the equation was first considered by Fermat, and a complete solution was given by Lagrange. By any name, the equation is an important one for several reasons – only some of which will be touched upon here – and its solution furnishes an ideal introduc tion to a whole branch of number theory, Diophantine Approximation . 1.2. First remarks on Pell’s equation. We call a solution ( x,y ) to (1) trivial if xy = 0. We always have at least two trivial solutions: ( x,y ) = ( ± 1 , 0), which we shall call trivial . As for any plane conic curve, as soon as there is one solution there are infinitely many rational solu tions ( x,y ) ∈ Q 2 , and all arise as follows: draw all lines through a single point, say ( − 1 , 0), with rational slope r , and calculate the second intersection point ( x r ,y r ) of this line with the quadratic equation (1). The above procedure generates all rational solutions and thus contains all in teger solutions, but figuring out which of the rational solutions are integral is not straightforward. This is a case where the question of integral solutions is essentially different, and more interesting, than the question of rational solutions. Henceforth when we speak of ‘solutions” ( x,y ) to (1) we shall mean integral solutions. Let us quickly dispose of some uninteresting cases. Proposition 1. If the Pell equation x 2 − dy 2 = 1 has nontrivial solutions, then d is a positive integer which is not a perfect square. Proof. • ( d = − 1): The equation x 2 + y 2 = 1 has four trivial solutions: ( ± 1 , 0) , (0 , ± 1). • ( d < − 1): Then x ̸ = 0 = ⇒ x 2 − dy 2 ≥ 2, so (1) has only the solutions ( ± 1 , 0). • ( d = N 2 ): Then x 2 − dy 2 = ( x + Ny )( x − Ny ) = 1, and this necessitates either: x + Ny = x − Ny = 1 in which case x = 1, y = 0; or x + Ny = x − Ny = − 1 , 1 2 THE PELL EQUATION in which case x = − 1 ,y = 0: there are only trivial solutions. From now on we assume that d is a positive integer which is not a square. Any nontrivial solution must have xy ̸ = 0. Such solutions come in quadruples: if ( x,y ) is any one solution, so is ( − x,y ), ( x, − y ) and ( − x, − y ). Let us therefore agree to restrict our attention to positive solutions : x,y ∈ Z + ....
View
Full Document
 Spring '11
 Staff
 Math, Number Theory, Irrational number, Pell, Pell equation

Click to edit the document details