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4400pellnotes

# 4400pellnotes - THE PELL EQUATION 1 Introduction Let d be a...

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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 (1611-1685). 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 + ....
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4400pellnotes - THE PELL EQUATION 1 Introduction Let d be a...

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