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

1. Jul 4, 2011

### RamaWolf

Let N be an integral non-square number, we have Pell's equation by

(1) x$^{2}$-N*y$^{2}$ = 1

associated with the 'negative Pell' by

(2) x$^{2}$-N*y$^{2}$ = -1

There are two well known theorems:

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

Therorem2: It is necessary (but not sufficient) for (2) to have a solution, that N = a$^{2}$ + b$^{2}$,
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 $\sqrt{N}$

Therorem3: Let (x$_{1}$,y$_{1}$) be a solution of (2), then x'$_{1}$ := 2*x$_{1}$$^{2}$+1 and y'$_{1}$:=2*x$_{1}$*y$_{1}$ are a solution of (1)