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Jacobson, Michael J. jun.; Williams, Hugh C.

The size of the fundamental solutions of consecutive Pell equations. (English) Zbl 0961.11007

Exp. Math. 9, No. 4, 631-640 (2000).
The authors look at the two Pellian equations: \(X^2-(D-1) Y^2=1\) and \(X^2-DY^2=1\). If \((X_0,Y_0)\) and \((X_1,Y_1)\) are least positive solutions, respectively of these Pell equations, then let \(\rho(D)= \log X_1/ \log X_0\). They show that \(\rho(D)\) can be arbitrarily large, and provide heuristic evidence for infinitely many values of \(D\) for which \(\rho(D)\gg \sqrt D\log\log D/ \log D\).
This is particularly important since, as the authors indicate, the result by F. Halter-Koch in Section 3 of [Abh. Math. Semin. Univ. Hamb. 59, 171-181 (1989; Zbl 0718.11054)] has an unrecoverable error, thereby invalidating that result.
The authors are able to go in a different direction to obtain a better result than Y. Yamamoto’s result given in [Osaka J. Math. 8, 261-270 (1971; Zbl 0243.12001)], which Halter-Koch’s result purported to advance. They indeed establish that there are infinitely many values of \(D\) such that \(\rho(D) \gg D^{1/6}/ \log D\).
Reviewer: Richard A.Mollin (Calgary)

Cited in 1 Document

MSC:

11D09 Quadratic and bilinear Diophantine equations
11R11 Quadratic extensions
11R27 Units and factorization
11Y40 Algebraic number theory computations

Keywords:

continued fractions; real quadratic field; Pell equations

Citations:

Zbl 0718.11054; Zbl 0243.12001

Software:

LiDIA
Cite Review PDF
Full Text: DOI Euclid EuDML

Online Encyclopedia of Integer Sequences:

Numbers n such that both n-1 and n are nonsquares and the least positive solutions to the Pell equations x1^2 - n*y1^2 =1 and x0^2-(n-1)*y0^2 = 1 have a record for rho(n)=log(x1)/log(x0).

References:

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