Difference between revisions of "Pell equation"
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Notice that if <math>D = d^2</math> is a perfect square, then this problem can be solved using [[difference of squares]]. We would have <math>x^2 - Dy^2 = (x+dy)(x-dy) = 1</math>, from which we can use casework to quickly determine the solutions. | Notice that if <math>D = d^2</math> is a perfect square, then this problem can be solved using [[difference of squares]]. We would have <math>x^2 - Dy^2 = (x+dy)(x-dy) = 1</math>, from which we can use casework to quickly determine the solutions. | ||
− | Alternatively, | + | Alternatively, if D is a nonsquare then there are infinitely many distinct solutions to the pell equation. To prove this it must first be shown that there is a single solution to the pell equation. |
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+ | Claim: If D is a positive integer that is not a perfect square, then the equation x^2-Dy^2 = 1 has a solution in positive integers. | ||
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+ | Proof: Let <math>c_{1}</math> be an integer greater than 1. We will show that there exists integers <math>t_{1}</math> and <math>w_{1}</math> such that <math>t_{1}-w_{1}\sqrt{D} < \frac{1}{c_{1}}</math> with <math>w_{1} \le c_{1}</math>. Consider the sequence <math>l_{k} = [k\sqrt{D}+1] \rightarrow 0 < l_{k}-k\sqrt{d} \le 1</math> <math>\forall</math> <math>0 \le k \le c_{1}</math>. By the pigeon hole principle it can be seen that there exists distinct integers i and j such that i < j and <math>\frac{p-1}{c_{1}} < l_{i}-i\sqrt{D} \le \frac{p}{c_{1}}, \frac{p-1}{c_{1}} < l_{j}-j\sqrt{D} \le \frac{p}{c_{1}}</math> for some positive integer <math>1 \le p \le c_{1}</math>. | ||
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== Family of solutions == | == Family of solutions == |
Revision as of 21:14, 21 January 2011
A Pell equation is a type of diophantine equation in the form for a natural number . Generally, is taken to be square-free, since otherwise we can "absorb" the largest square factor into by setting .
Notice that if is a perfect square, then this problem can be solved using difference of squares. We would have , from which we can use casework to quickly determine the solutions.
Alternatively, if D is a nonsquare then there are infinitely many distinct solutions to the pell equation. To prove this it must first be shown that there is a single solution to the pell equation.
Claim: If D is a positive integer that is not a perfect square, then the equation x^2-Dy^2 = 1 has a solution in positive integers.
Proof: Let be an integer greater than 1. We will show that there exists integers and such that with . Consider the sequence . By the pigeon hole principle it can be seen that there exists distinct integers i and j such that i < j and for some positive integer .
Family of solutions
Given a smallest solution , then all solutions are of the form for natural numbers .
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Continued fractions
The solutions to the Pell equation when is not a perfect square are connected to the continued fraction expansion of . If is the period of the continued fraction and is the th convergent, all solutions to the Pell equation are in the form for positive integer .
Generalization
A Pell-like equation is a diophantine equation of the form , where is a natural number and is an integer.