JP Journal of Algebra, Number Theory and Applications
Volume 4, Issue 1, Pages 159 - 207
(April 2004)
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A
CONTINUED FRACTION APPROACH TO THE DIOPHANTINE
EQUATION ax2
– by2
= ±1
R. A. Mollin (Canada)
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Abstract: We
revisit the Diophantine equation of the title,
and related equations, from new perspectives
that add connections to continued fractions,
fundamental units of real quadratic fields,
Jacobi symbol equations, and ideal theory. We
also develop an analogous theory for the
related equation
with
Included in both cases is a means for finding
the fundamental unit of the underlying
quadratic order "halfway" along the
period of the simple continued fraction
expansion of
where
We show, as well, how the fundamental units of
these two orders may be linked explicitly in
the two simple continued fraction expansions.
In addition, we explore the links with the
solvability of the norm form equation
Moreover, we give explicit necessary and
sufficient conditions for the parity of the
period length of the simple continued fraction
expansion of
to be even in terms of the solvability of
the title equation plus a related such
equation, and an analogue for
Also, we give a criterion for the period
length to be odd in terms of ambiguous ideal
classes. |
Keywords and phrases: quadratic
diophantine equation, continued fractions, norm form
equations, fundamental units. |
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