Difference between revisions of "Pell equation"
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− | A '''Pell equation''' is a type of [[diophantine equation]] in the form <math>x^2-Dy^2 = | + | A '''Pell equation''' is a type of [[diophantine equation]] in the form <math>x^2-Dy^2 = \pm1</math> for a [[natural number]] <math>D</math>. Generally, <math>D</math> is taken to be square-free, since otherwise we can "absorb" the largest square factor <math>d^2 | D</math> into <math>y</math> by setting <math>y' = dy</math>. |
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. |
Revision as of 18:22, 27 August 2016
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 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 i, j, and p such that i < j, and
.
So we now have
.
We can now create a sequence of such that and which implies r and s. However we can see by the pigeon hole principle that there is another infinite sequence which will be denoted by such that . Once again, from the pigeon hole principle we can see that there exist integers f and g such that mod H, mod H, and . Define and notice that . Also note that mod H which means that Y = 0 mod H also. We can now see that is a nontrivial solution to pell's equation.
Contents
[hide]Family of solutions
Let be the minimal solution to the equation . Note that if are solutions to this equation then which means is another solution. From this we can guess that is obtained from . This does indeed generate all the solutions to this equation. Assume there was another solution . Then there exists some m such that
.
However, it can be seen that
Meaning is a solution smaller than the minimal solution which is a contradiction. This article is a stub. Help us out by expanding it.
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.
Introductory Problems
If and are the solutions to the equation , then .