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x*x-D*y*y=1, and I have in particular looked at the case D=n*n-3 which

contains solutions with high values for x and y, such as for D=61.

My simple studies have led me to formulate the following conjecture:

When D in Pell’s equation x*x-D*y*y=1 is

1) of the form n*n-3, and

2) a prime number

then

x+1 contains a set of factors which

a) includes D and

b) includes one or more factors from y twice, i.e. in a squared form.

For instance: x and y are the solution of x2-397y2=1 [x=838 721 786

045 180 184 649, y= 42 094 239 791 738 438 433 660] and

X+1 has the factors 2, 5^2, 17^2, 37^2, 173^2, 397, 1889^2, and

y the factors 2^2, 3^3, 5, 17, 37, 173, 383, 1 889, 990 151, of

which 5 appear as squares in x.

My idea is that if z=x+1 and (z-1)*(z-1)-Dy=1, then z*z-2z-Dy=0 and it

is possible to eliminate both D and more factors from the terms in the

equation, thereby revealing simpler relations between smaller terms.

(I also nourish a hope to find a general solution which will give me

the lowest solution, in additions to the other techniques that are

reported).

Is this something that you can prove false, or is it correct, perhaps

a well known fact?

Thank you for any comments you might have.