Let N be an integral non-square number, we have Pell's equation by(adsbygoogle = window.adsbygoogle || []).push({});

(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)

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

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