# Pell equation

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

(1) |

as well as the more general equation

(2) |

where is a positive integer, is an irrational number, is an integer, and the unknowns and are integers.

If , are the convergent fractions for the expansion of in a continued fraction with period , then the positive solutions to (1) take the form

where is any natural number such that is even.

All the solutions to (1) are derived from the formula

where is any integer and 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 , solutions exist if and only if is odd. For , (2) always has solutions. The solutions to the Pell equation for are used in determining the units of the quadratic field . The solutions to a Pell equation are used to determine automorphisms of a binary quadratic form ; these enable one to use one solution to the Diophantine equation 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=11604