Topology of Numbers · Problem 4.3.1

Classification of Quadratic Forms — Allen Hatcher

Mathematics — Number Theory · pells-equation

Problem

For the quadratic form x2−14y2x^2 - 14y^2 do the following things: (a) Draw the separator line in the topograph and compute the continued fraction for 14\sqrt{14}. (b) Find the smallest positive integer solutions of x2−14y2=1x^2 - 14y^2 = 1 and x2−14y2=−1x^2 - 14y^2 = -1, if these equations have integer solutions. (c) Find the linear fractional transformation that gives the periodicity translation along the separator line and use this to find a second positive solution of x2−14y2=1x^2 - 14y^2 = 1. (d) Determine the integers nn with ∣n∣≤12|n| \leq 12 such that the equation x2−14y2=nx^2 - 14y^2 = n has an integer solution. (Do not forget the possibility that there could be solutions (x,y)(x, y) that are not primitive.)

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