From: John Robertson <jpr2718@aol.com>
Subject: Re: Size of Fundamental Solution to Pell Equation
Date: Sat, 27 Jan 2001 18:09:12 GMT
Newsgroups: sci.math

On December 24, 2000, I wrote:

> Let t, u be the least positive solution to x^2 - dy^2 = 4.
> Let e=(t + u*sqrt(d))/2.  Schur, Gottingen Nachrichter, 1918,
> pp. 30-36, shows e < d^d^(1/2).  Loo-Keng Hua, Bulletin of
> the AMS, vol 48, 1942, p 731, shows for d == 0 or 1 (mod 4)
> that log e < (d^(1/2))*((1/2) log d + 1).
>
>  Has this bound been improved?  Is it the best possible?

Gary Walsh has replied that these bounds have not been improved, and are
conjectured to be the best possible.  This question is tied to the
question of whether there are infinitely many real quadratic fields with
class number 1.  If there are, then the cited bounds cannot be improved.
 Richard A. Mollin, Quadratics, CRC Press, Boca Raton, 1996, page 172,
discusses these issues and cites further references, including C. L.
Siegel, Uber die Klassenzahl quadratischer Zahlkorper, Acta Arith., Vol
1, 1935, pages 83-86.

John Robertson


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