Keywords
., Continued fraction; Convergent; Fundamental solution; Indefinite binary quadratic form; Quadratic diophantine equation
Extending theorems of Serret and Pavone
Abstract
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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.
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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