Namespaces
Variants
Actions

Difference between revisions of "Pell equation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
A Diophantine equation (cf. [[Diophantine equations|Diophantine equations]]) of the form
 
A Diophantine equation (cf. [[Diophantine equations|Diophantine equations]]) of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p0719801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$x^2-dy^2=1,\label{1}$$
  
 
as well as the more general equation
 
as well as the more general equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p0719802.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$x^2-dy^2=c,\label{2}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p0719803.png" /> is a positive integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p0719804.png" /> is an irrational number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p0719805.png" /> is an integer, and the unknowns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p0719806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p0719807.png" /> are integers.
+
where $d$ is a positive integer, $\sqrt d$ is an irrational number, $c$ is an integer, and the unknowns $x$ and $y$ are integers.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p0719808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p0719809.png" /> are the convergent fractions for the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198010.png" /> in a [[Continued fraction|continued fraction]] with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198011.png" />, then the positive solutions to (1) take the form
+
If $P_s/Q_s$, $s=0,1,\ldots,$ are the convergent fractions for the expansion of $\sqrt d$ in a [[Continued fraction|continued fraction]] with period $k$, then the positive solutions to \ref{1} take the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198012.png" /></td> </tr></table>
+
$$x=P_{kn-1},\quad y=Q_{kn-1},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198013.png" /> is any natural number such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198014.png" /> is even.
+
where $n$ is any natural number such that $kn$ is even.
  
All the solutions to (1) are derived from the formula
+
All the solutions to \ref{1} are derived from the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198015.png" /></td> </tr></table>
+
$$x+y\sqrt d=\pm(x_0+y_0\sqrt d)^n,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198016.png" /> is any integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198017.png" /> is the solution with the least positive values for the unknowns. The general equation (2) either has no solutions at all or has infinitely many. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198018.png" />, solutions exist if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198019.png" /> is odd. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198020.png" />, (2) always has solutions. The solutions to the Pell equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198021.png" /> are used in determining the units of the [[Quadratic field|quadratic field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198022.png" />. The solutions to a Pell equation are used to determine automorphisms of a binary [[Quadratic form|quadratic form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198023.png" />; these enable one to use one solution to the Diophantine equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071980/p07198024.png" /> to obtain an infinite set of solutions.
+
where $n$ is any integer and $(x_0,y_0)$ is the solution with the least positive values for the unknowns. The general equation \ref{2} either has no solutions at all or has infinitely many. For $c=-1$, solutions exist if and only if $k$ is odd. For $c=4$, \ref{2} always has solutions. The solutions to the Pell equation for $c=\pm1,\pm4$ are used in determining the units of the [[Quadratic field|quadratic field]] $R(\sqrt d)$. The solutions to a Pell equation are used to determine automorphisms of a binary [[Quadratic form|quadratic form]] $Ax^2+Bxy+Cy^2$; these enable one to use one solution to the Diophantine equation $Ax^2+Bxy+Cy^2=n$ to obtain an infinite set of solutions.
  
 
Equation (1) was examined by W. Brouncker (1657), P. Fermat and J. Wallis. L. Euler, on account of a misunderstanding, ascribed it to J. Pell.
 
Equation (1) was examined by W. Brouncker (1657), P. Fermat and J. Wallis. L. Euler, on account of a misunderstanding, ascribed it to J. Pell.

Latest revision as of 09:21, 27 April 2014

A Diophantine equation (cf. Diophantine equations) of the form

$$x^2-dy^2=1,\label{1}$$

as well as the more general equation

$$x^2-dy^2=c,\label{2}$$

where $d$ is a positive integer, $\sqrt d$ is an irrational number, $c$ is an integer, and the unknowns $x$ and $y$ are integers.

If $P_s/Q_s$, $s=0,1,\ldots,$ are the convergent fractions for the expansion of $\sqrt d$ in a continued fraction with period $k$, then the positive solutions to \ref{1} take the form

$$x=P_{kn-1},\quad y=Q_{kn-1},$$

where $n$ is any natural number such that $kn$ is even.

All the solutions to \ref{1} are derived from the formula

$$x+y\sqrt d=\pm(x_0+y_0\sqrt d)^n,$$

where $n$ is any integer and $(x_0,y_0)$ is the solution with the least positive values for the unknowns. The general equation \ref{2} either has no solutions at all or has infinitely many. For $c=-1$, solutions exist if and only if $k$ is odd. For $c=4$, \ref{2} always has solutions. The solutions to the Pell equation for $c=\pm1,\pm4$ are used in determining the units of the quadratic field $R(\sqrt d)$. The solutions to a Pell equation are used to determine automorphisms of a binary quadratic form $Ax^2+Bxy+Cy^2$; these enable one to use one solution to the Diophantine equation $Ax^2+Bxy+Cy^2=n$ to obtain an infinite set of solutions.

Equation (1) was examined by W. Brouncker (1657), P. Fermat and J. Wallis. L. Euler, on account of a misunderstanding, ascribed it to J. Pell.

References

[1] A.Z. Walfisz, "Pell's equation" , Tbilisi (1952) (In Russian)
[2] A.D. Gel'fond, "The solution of equations in integers" , Noordhoff (1960) (Translated from Russian)
[3] W.J. Leveque, "Topics in number theory" , 1 , Addison-Wesley (1965)


Comments

References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Clarendon Press (1979)
How to Cite This Entry:
Pell equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pell_equation&oldid=31941
This article was adapted from an original article by A.A. Bukhshtab (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article