Journal of Integer Sequences, Vol. 20 (2017), Article 17.10.5

Extending Theorems of Serret and Pavone

Keith R. Matthews
School of Mathematics and Physics
University of Queensland
Brisbane 4072

John P. Robertson
Actuarial and Economic Services Division
National Council on Compensation Insurance
Boca Raton, FL 33487

Anitha Srinivasan
Department of Mathematics
Saint Louis University-Madrid Campus
Avenida del Valle 34
28003 Madrid


We extend theorems of Serret and Pavone for solving f(x,y) = ax2 + bxy + cy2 = μ, where a > 0, gcd(x, y) = 1, y > 0. Here d = b2 - 4ac > 0 is not a perfect square and $0 < |μ| < √d/2. If μ > 0, Serret proved that x/y is a convergent to ρ = (-b + √d)/2a or σ = (-b - √d)/2a. If μ < 0, we are able to modify Pavone's approach and show that with at most one exception, the solutions are convergents to ρ or σ.

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Received July 18 2017; revised version received September 16 2017. Published in Journal of Integer Sequences, October 29 2017.

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