about that green line...
Markoff numbers are positive integers that appear
in solutions to the equation
a^{2} + b^{2} + c^{2} = 3abc .
The Markoff numbers can be cataloged in a tree as shown below
(and described on the previous page).
All tree vertices in this diagram are located on a grid
whose y coordinates are noted
in blue.
Each level of the tree has twice as many vertices as the previous level.
The first level's vertex is located at y = 1/2, the next level's
vertices are at 1/4 and 3/4, the next are at 1/8, 3/8, 5/8 and 7/8, and so on.
My purpose in establishing this coordinate system is to
note where the largest number in each level of the tree appears,
specifically to the right of the vertex
whose y coordinate is closest to 2/3.
Note that the numbers at the vertices closest
to the green line
(5, 29, 433, 37666, and 48928105) are the largest
numbers in their respective levels of the tree. (1 and 2 are special
cases and are not properly associated with any vertex in the tree.)
The tree can be extended indefinitely; the
largest number in any level always appears at the vertex
closest to the 2/3 coordinate.
Tom Ace
return to Tom's main Markoff number page
return to Tom's home page
