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The Pellians: Pell's equation and its twin, the negative Pellian

  1. Jul 4, 2011 #1
    Let N be an integral non-square number, we have Pell's equation by

    (1) x[itex]^{2}[/itex]-N*y[itex]^{2}[/itex] = 1

    associated with the 'negative Pell' by

    (2) x[itex]^{2}[/itex]-N*y[itex]^{2}[/itex] = -1

    There are two well known theorems:

    Therorem1: (1) has infinitely many solutions, depending on the continued fraction expansion of [itex]\sqrt{N}[/itex]

    Therorem2: It is necessary (but not sufficient) for (2) to have a solution, that N = a[itex]^{2}[/itex] + b[itex]^{2}[/itex],
    for some integral co-prime a and b (i.e. a and b have no common divisior); if (2) has a solution,
    there are infinitely many, depending on the continued fraction expansion of [itex]\sqrt{N}[/itex]

    Therorem3: Let (x[itex]_{1}[/itex],y[itex]_{1}[/itex]) be a solution of (2), then x'[itex]_{1}[/itex] := 2*x[itex]_{1}[/itex][itex]^{2}[/itex]+1 and y'[itex]_{1}[/itex]:=2*x[itex]_{1}[/itex]*y[itex]_{1}[/itex] are a solution of (1)
     
  2. jcsd
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