A Markoff number (named after
Andrei A. Markoff) is a number that appears in a positive integer
solution to the equation
a
In the example shown, 7561 = 3 x 194 x 13 - 5. Note also that 3ab-c = (a ^{2} + b^{2}) / c
(this follows from the Markoff equation a ^{2} + b^{2} + c^{2} = 3abc ).
For example, 7561 = (194 ^{2} + 13^{2}) / 5.
The tree can be continued indefinitely. The numbers in regions adjacent to the 1 region in the tree as shown above (2, 5, 13, 34, 89, 233, ...) are alternate Fibonacci numbers. Numbers in regions adjacent to the 2 region (1, 5, 29, 169, 985, 5741, ...) are alternate Pell numbers. Markoff proved that this form of tree will generate all possible Markoff numbers. A well-known conjecture states that no number appears in two places in the tree, which is tantamount to saying that no number c can be the largest number in each of two different triples (a _{1},b_{1},c) and
(a_{2},b_{2},c)
of positive integers that satisfy the Markoff equation. This is known
as the unicity (uniqueness) conjecture for the Markoff numbers.
Baragar and
Button have published proofs of
uniqueness for certain specific subsets of the Markoff numbers,
but the general unicity conjecture remains unproven--although
a faulty proof was published
by Gerhard Rosenberger in 1976, and another
claimed proof by Qing Zhou was withdrawn.
Markoff numbers play a role in (among other things) the theory of rational approximation of irrational numbers, but such applications are beyond the scope of these web pages. My purpose here is to describe a few interesting properties of the Markoff numbers, with an emphasis on material that doesn't require advanced math knowledge to appreciate. More about the Markoff numbers: - calculating Markoff numbers with matrices (updated 4 Jan 2005)
- what's with that green line in the tree?
- references
- More to come; please check back.
The Book of Numbers which introduced me to (among other things)
the Markoff numbers. The form of the tree above follows the example
on p. 188 of their book.
Markoff's name isn't spelled consistently in English language mathematical literature, with some authors instead preferring "Markov", "Markov number", "Markov numbers", and "Markov equation". Tom Ace return to Tom's home page |