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Lagrange's Algorithm Revisited: Solving ***a**t*^{2} + *btu* + *c**u*^{2} = *n* in the Case of
Negative Discriminant

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Keith R. Matthews

School of Mathematics and Physics

University of Queensland

Brisbane, QLD 4072

Australia

and

Centre for Mathematics and its Applications

Australian National University

Canberra, ACT 0200

Australia

**Abstract:**

We make more accessible a neglected continued fraction algorithm of
Lagrange for solving the equation
*a**t*^{2} + *btu* + *c**u*^{2} = *n*
in relatively
prime integers *t*, *u*, where *a* > 0,
gcd(*a*,*n*) = 1, and *D* = *b*^{2} - 4*ac*
< 0.
The cases *D* = -4 and *D* = -3 present a
consecutive convergents phenomenon which aids
the search for solutions.

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Received
August 26 2014; revised version received September 22 2014.
Published in *Journal of Integer Sequences*, November 5 2014.

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