From: Jpr2718@aol.com (John Robertson)
Subject: Periods of Continued Fractions
Date: 12 May 00 12:25:16 GMT
Newsgroups: sci.math.numberthy
Summary: [missing]

Alf van der Poorten pointed me to papers that answered my final
queries regarding lengths of periods of continued fractions.

Kenneth S. Williams and Nicholas Buck, "Comparison of the Lengths of
the Continued Fractions of sqrt(D) and (1+sqrt(D))/2," Proceedings of
the AMS, v 120, no 4, April 1994, pp 995-1002.

Noburo Ishii, Pierre Kaplan, and Kenneth S. Williams, "On Eisenstein's
problem," Acta Arithmetica, LIV (1990), pp 323-345.

A. J. van der Poorten, "Fractions of the Period of the Continued
Fraction Expansion of Quadratic Integers," Bulletin of the Australian
Mathematical Society, v 44 (1991), pp 155-169.

Pierre Kaplan and Kenneth S. Williams, "Pell's Equations X^2 –
mY^2 = -1, -4 and Continued Fractions," Journal of Number Theory, v 23
(1986), pp 169-182.

To summarize, fix a non-square positive integer D == 1 (mod 4), D >=
17, let L_1 be the length of the period of the cf expansion of
sqrt(D), and let L_4 that for (1+sqrt(D))/2.

The following two facts are well known.  L_1 == L_4 (mod 2), so
x^2-Dy^2=-1 (hereafter, the -1 equation) is solvable iff the -4
equation is solvable (not necessarily in odd integers).  If the -4
equation is solvable, then the +4 equation is solvable in odd integers
iff the -4 equation is solvable in odd integers.

From  papers of Williams and Buck, and of Ishii, Kaplan and Williams,
we have

If L_1 == 1 (mod 2) then
either
        L_1 == L_4 (mod 4), and
        L_4 + 4 <= L_1 <= 5L_4, and
        +4 and -4 equations have solutions
        in odd integers,
or
        L_1 + L_4 == 0 (mod 4), and
        (1/3)L_4 <= L_1 <= 3L_4 - 4, and
        +4 and -4 equations do not have solutions
        in odd integers.

If L_1 == 0 (mod 2) then L_1 == L_4 (mod 4) and
either
        L_4 + 4 <= L_1 <= 5L_4, and
        +4 equation has solutions in odd integers,
or
        (1/3)L_4 <= L_1 <= 3L_4 - 8, and
        +4 equation does not have solutions
        in odd integers.

John Robertson