Journal of Integer Sequences, Vol. 17 (2014), Article 14.11.1

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
Centre for Mathematics and its Applications
Australian National University
Canberra, ACT 0200


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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