about that green line...

Markoff numbers are positive integers that appear in solutions to the equation a2 + b2 + c2 = 3abc . The Markoff numbers can be cataloged in a tree as shown below (and described on the previous page).
Markoff numbers
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

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