Pell equation

For a nonsquare positive integer $D$, the Pell equation isPeriodic continued fractions of $\sqrt{D}$ produce infinitely many integer solutions.

For $x^2-D{y}^2=1$ with nonsquare positive integer $D$, the solution with $x>1$, $y>0$ and smallest $x$ is the fundamental solution. It generates other positive solutions through powers of $x+y\sqrt{D}$ and can be found using continued fraction convergents of $\sqrt{D}$.

The Diophantine equation $x^2-D{y}^2=-1$, usually with a nonsquare positive integer $D$ and integer unknowns. It need not have a solution. For $D=2$, multiplying $u+v\sqrt{2}$ by $3+2\sqrt{2}$ preserves norm $-1$ and gives $(u,v)\mapsto (3u+4v,2u+3v)$.

In a real quadratic integer ring, multiplying by a positive norm-one fundamental unit preserves a prescribed norm. Choose an exponent so that a positive representative lies in a half-open interval $[A,A\epsilon )$. Its conjugate is the prescribed norm divided by the representative, so both integer coefficients are bounded on that interval. Enumerating these reduced representatives gives a complete finite list of unit orbits of solutions.

A generalized Pell equation is $x^2-d{y}^2=N$ for a fixed nonsquare positive integer $d$ and nonzero integer $N$. Multiplication by units of a real quadratic number field preserves the norm up to sign, reducing solutions to finitely many unit orbits when an appropriate bounded reduction interval is used.