Lagrange's Algorithm Revisited: Solving at2 + btu + cu2 = n in the Case of
Negative Discriminant
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
at2 + btu + cu2 = n
in relatively
prime integers t, u, where a > 0,
gcd(a,n) = 1, and D = b2 - 4ac
< 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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