Pell's equation

As Pellsche equation (after John Pell, 1611-1685 ) refers to a Diophantine equation of the form

With positive integer.

Is a square number, then the equation has apparently only the trivial solutions. Otherwise, there are infinitely many solutions, which can be determined by means of the continued fraction expansion of. The related equations and are often also called Pellsche equations.

The equation is falsely attributed to John Pell. Correct would be the name of Fermat's equation.

Algebraic Number Theory

Finding all solutions for specific equivalent to finding the units of wholeness ring of real quadratic number field. After the Dirichlet Unit Theorem the unit group has rank 1, that is, there is a fundamental unit (or base unit ) with which make all solutions and represent.

Solutions

Solution by means of the continued fraction expansion

The continued fraction expansion of a quadratic irrational number is infinite and periodic. For example, the continued fraction

If you break the development of each of the Agent, is obtained starting with

And finds at the sites and the solutions

Of and

Of.

Next, it is found that for each element of the aborted continued fraction expansion of length is a solution of a Pell equation with right-hand side.

Generate additional solutions based on a known

If a solution is known, can be other solutions with a matrix multiplication determined. It is

The Pellsche equation has the minimal solution. The obvious solutions are then obtained to be

Etc.

The cattle problem of Archimedes

The solution to the cattle problem of Archimedes one encounters (if you sent included) on the Pellsche equation for parameters as the minimum solution

Has. For the cattle problem need not, however, the minimal solution, but a (more precisely, the smallest ) solution, in which a multiple of being.

Alternatively, you can search for Pellsche equation with parameters, the minimal solution (without constraint now ), which is of the following order (see above source):

( Not coincidentally, is 2 · 3.7653 · 10,103,272 ≈ (2 · 1.0993199 · 1044 ) in 2329, whereby the relationship between the minimal solutions of the two Pell equations is numerically established.)

For the cattle problem itself is as an intermediate result, the number 4657 957 · · · y2 ≈ 1.5401 10,206,537 of concern. The end result is the 50,389,082 -fold it, so about 7,760 · 10,206,544th