Pell Equation. IV. Fastest algorithm for solving the Pell equation

概要

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The fastest algorithm for solving the Pell equations, x^2-Dy^2=1 (called Pell-1) and x^2-Dy^2=-1 (Llep-1) , are demonstrated with two typical examples. The essence of the algorithm is i) to obtain the periodic continued fraction expression for the square root of D, ii) to prepare four caterpillar graphs by using the terms derived above, and iii) to set a 3×3(for Pell) or 2×2(for Llep) determinant whose elements are the topological indices (Z’s) of those graphs, and iv) to calculate the determinant. The dramatic shortening of the procedure comes from the finding that the continuant is equivalent to the topological index of the caterpillar graph directly derived from the continued fraction expansion of the square root of D.
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