Pell equation

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)

References

[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Clarendon Press (1979)