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A simple proof of the Markoff conjecture for prime powers. (English) Zbl 1133.11023

The authors give a markedly geometric proof of the known arithmetic result that the Markoff equation \(x^2+ y^2+z^2 = 3 x y z\) has at most one solution in positive integers \((x,y,z)\) with \(x\leq y \leq z\) for any given prime power \(z\). The geometry of the proof comes from H. Cohn’s 1955 observation that the tree of solutions to the above Markoff equation corresponds one-to-one to the simple closed geodesics of the once-punctured hyperbolic torus uniformized by the commutator subgroup of \(\text{PSL}(2, \mathbb Z)\). Solutions sharing largest value \(z\) correspond to geodesics of equal depth of penetration into the cusp; there is thus a parabolic translation that takes a lift to the upper half-plane of one geodesic to the other. This leads to a congruence that is easily used in this prime power case to show uniqueness.

MSC:

11D25 Cubic and quartic Diophantine equations
11J06 Markov and Lagrange spectra and generalizations
11J70 Continued fractions and generalizations
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References:

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