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Positive integer solutions to $(x+y)^2+(y+z)^2+(z+x)^2=12xyz$
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abstract
In this paper, we give a specific way of describing positive integer solutions of a Diophantine equation $(x+y)^2+(y+z)^2+(z+x)^2=12xyz$ and introduce a generalized cluster pattern behind it.
Forward citations
Cited by 2 Pith papers
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Open surfaces with a triangle at infinity
Triangle surfaces are exactly the Markov-type cubics xyz=x^2+y^2+z^2+ax+by+cz+d, and their automorphism groups are Gσ⋊Γ with Γ one of five finite groups from Table 1.
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Divisibility by $p$ for Markoff-like Surfaces
For most parameters with nonzero 3+a1+a2+a3 and p>=5, every nontrivial orbit on the generalized Markov surface has size divisible by p, with quadratic obstructions giving at least two or four orbits in special cases.
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