Re: Ed Barbeau's "Pell's Equation"


Karl M. Bunday wrote:
> sttscitrans@xxxxxxxxx wrote, replying to an earlier reply to his
> thread-opening question:
>
> >>|> A reviewer mentions that there is a chapter
> >>|> on "the cubic analog of Pell's equation"
> >>|> in Barbeau's book. Could anyone who has
> >>|> read this chapter tell me what this
> >>|> analog actually is ?
>
> > Thanks, I suspect the analog is
> > probably x^3 +ky^3 +(k^2)z^3 -3kxyz=1
>
> That appears to be what Barbeau had in mind, based on Power Play page
> 91, where the equation is given as
>
> "The cubic analogue of Pell's equation is
>
> x^3 + dy^3 + d^2z^3 - 3dxyz = 1
>
> If (x, y, z) is a solution in large positive integers, then from the
> factorization of the left side, x + d^(1/3)y + d^(2/3)z would be large
> and of x - d^(1/3)y, y - d^(1/3)z and x - d^(2/3)z would have to be very
> small. Thus we look for x/y and y/z to be close to the cube root of d."

Yes, x:y:z must be a good approximation to
d^(2/3):d^(1/3):1

> > The reviewer seems to suggest that Barbeau
> > gives a method of solving these higher
> > degree Pell analogs and as I have recently
> > found a method that seems to work, I
> > was wondering what Barbeau's method was.
> > Apparently, he also deals with Pell analogs
> > in his book "Powerplay", however according
> > to the inter-library loan service there isn't
> > a single copy available in the GB.
>
> That makes me feel glad to have Power Play in my home library.
>
> Barbeau, Edward J. (1997). Power Play. Washington, DC: Mathematical
> Association of America.
>
> Barbeau outlines a procedure which he describes on page 92 like this:
>
> "Unlike the quadratic case, the process for the cubic does not seem to
> have nice structural properties. While it might not work for every value
> of d, it seems to generate a solution surprisingly often. When it does
> generate a solution, it does not pick up every solution, but actually
> misses quite a few.

Yes, that sounds very familiar.


> The appendix describes the process in more detail, and begins by saying
> it is for persons with "university level mathematics."
>
> It's late in my time zone and I need to turn in, but I could type up
> more by request. I dare not paraphrase as I hardly understand this
> myself, having only secondary level mathematics except for some
> recreational reading I've done in the last few years.

I would be very grateful if you could type
out the main sections of Barbeau's method .
As his book does not seem to be that widely
available, I'm sure others would be interested too.


> > Perhaps, I'll have more luck with "Pell's Equation".
>
> I think that is the more recent and more thorough book by Barbeau on
> that subject. Good luck.

Thanks, who says sci.math can't be
informative.

.