Markoff-Rosenberger triples in arithmetic progression

Markoff–rosenberger Triples in Geometric Progression

We study solutions of the Markoff–Rosenberger equation ax + by + cz = dxyz whose coordinates belong to the ring of integers of a number field and form a geometric progression.

On the Uniqueness Conjecture for Markoff Triples

ν(θ) = inf{c : |θ − p/q| < c/q for infinitely many reduced fractions p/q}. The set of values {νi} of the Markoff function in the range ν(θ) > 1/3 is called the Markoff spectrum [7], which is a denumerable infinite set and νi ↓ 1/3. An other definition of the Markoff spectrum uses the integer solutions of the Diophantine equation x + y + z = 3xyz. (1) A solution (x, y, z) to this equation with 0...

A Simple Proof of the Markoff Conjecture for Prime Powers

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 ≤ c holds whenever c is a prime power.

On Triples in Arithmetic Progression

Avoiding triples in arithmetic progression ∗

Some patterns cannot be avoided ad infinitum. A well-known example of such a pattern is an arithmetic progression in partitions of natural numbers. We observed that in order to avoid arithmetic progressions, other patterns emerge. A visualization is presented that reveals these patterns. We capitalize on the observed patterns by constructing techniques to avoid arithmetic progressions. More for...