Pell ’ s Equation
نویسنده
چکیده
An arbitrary quadratic diophantine equation with two unknowns can be reduced to a Pell-type equation. How can such equations be solved? Recall that the general solution of a linear diophantine equation is a linear function of some parameters. This does not happen with general quadratic diophantine equations. However, as we will see later, in the case of such equations with two unknowns there still is a relatively simple formula describing the general solution. Why does the definition of Pell’s equations assume d is not a square? Well, for d = c2, c ∈ Z, the equation x2 −dy2 = a can be factored as (x− cy)(x+ cy) = a and therefore solved without using any further theory. So, unless noted otherwise, d will always be assumed not to be a square. The equation x2 −dy2 = a can still be factored as
منابع مشابه
Pell ’ s equation
1 On the so–called Pell–Fermat equation 2 1.1 Examples of simple continued fractions . . . . . . . . . . . . . 2 1.2 Existence of integer solutions . . . . . . . . . . . . . . . . . . 5 1.3 All integer solutions . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 On the group of units of Z[ √ D] . . . . . . . . . . . . . . . . . 8 1.5 Connection with rational approximation . . . . . . . . . . ....
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