Article

An elementary proof of uniqueness of Markoff numbers which are prime powers

07/2006;
Source: arXiv

ABSTRACT

We present a very elementary proof of the uniqueness of Markoff numbers which are prime powers or twice prime powers, in the sense that it uses neither algebraic number theory nor hyperbolic geometry.

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Available from: Ying Zhang, Aug 27, 2014
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• "A simple, short proof using the hyperbolic geometry of the modular torus as used by Cohn in [5] has been obtained a bit later but only recently posted by Lang and Tan [10]. See [15] for a completely elementary proof which uses neither hyperbolic geometry nor algebraic number theory. A stronger result along the same lines has been obtained by Button in [3]; in particular, the Markoff numbers which are " small " (≤ 10 35 ) multiples of prime powers are unique. "
Article: Congruence and Uniqueness of Certain Markoff Numbers
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ABSTRACT: By making use of only simple facts about congruence, we first show that every even Markoff number is congruent to 2 modulo 32, and then, generalizing an earlier result of Baragar, establish the uniqueness for those Markoff numbers c where one of 3c - 2 and 3c + 2 is a prime power, 4 times a prime power, or 8 times a prime power.
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Article: A simple proof of the Markoff conjecture for prime powers
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ABSTRACT: We give a simple and independent proof of the result of Jack Button and Paul Schmutz that the Markoff conjecture on the uniqueness of the Markoff triples (a,b,c), where a, b, and c are in increasing order, holds whenever $c$ is a prime power.
Preview · Article · Sep 2005 · Geometriae Dedicata