# Pell equation

a diophantische equation of the form

< math> x^2 - dy^2= \ pm 1< /math>

for positively integral [itex] D< /math> Pell equation is called (after John Pell).

Is [itex] D< /math> a Quadratzahl, then possesses the equation obviously only the trivial solutions [itex] (\ pm 1, 0)< /math> (and [itex] (0, \ pm 1)< /math> for [itex] D = 1< /math>). Otherwise there are infinitely many solutions, which one sqrt with the continued fraction development <of> math \ {D}< /math> to determine can.

## algebraic number theory

finding all solutions is for special [itex] D< /math> equivalent in addition, the units of the entireness ring of [itex] \ mathbf {Q} (\ {D sqrt})< /math> to find. After the Dirichlet unit set the group of units has rank 1, and so there is a fundamental unit [itex] \ varepsilon = x_0 + \ sqrt {D} y_0< /math>, so that itself all solutions as [itex] \ pm \ varepsilon^n, n \ in \ mathbf {Z}< /math> to represent leave.

## the cattle problem of the Archimedes

with the solution of the cattle problem of the Archimedes one encounters x^2 the Pell <equation> math - 4729494y^2 = 1< /math> with minimum solution [itex] (109931986732829734979866232821433543901088049, 50549485234315033074477819735540408986340)< /math>, if one counts skillfully - otherwise one will have to solve a Pell equation with much larger minimum solution.