I read an interesting article titled A Triplet Tree Forms One of the Most Beautiful Structures in Math in Quanta Magazine that begins with the intriguing statement:
The Markov numbers reveal the secrets of irrational numbers and the patterns of the Fibonacci sequence. But there’s one question about them that has resisted proof for over a century.
The statement is followed by the diagram and caption shown in Figure 1.
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This tree shows how you can uniquely determine a Markov triple. Start from the largest number in your triple. Then cross two adjacent branches to move down the tree. For example, if you start from 194, you will reach 13 and 5. |
Most people are only familiar with a handful of numbers that can’t be written as fractions, like \( \sqrt{2} \) or \( \pi \). But such numbers, called irrational numbers, are far more plentiful than fractions or rational numbers. How easy are they to approximate with fractions? If you use a fraction with an arbitrarily large denominator, you can get arbitrarily close. As is well known, 22/7 gives a decent approximation of \( \pi \), 355/113 is even better. But some irrational numbers are harder to approximate than others, meaning that you need to use a very big denominator to get a close approximation. The very toughest turns out to be the golden ratio \( \phi = (1+\sqrt{5})/2 \). It is, in a specific mathematical sense, the number that is “farthest” from being rational.What’s the next-farthest? And the next? The sequence of tough-to-approximate irrational numbers turns out to be given by the integer solutions to a deceptively simple equation that has no obvious connection to approximating irrational numbers. This connection was proved by Andrey Markov, a prescient Russian mathematician, in 1879. Markov is famous for coming up with a concept in probability theory called Markov chains, which are used in everything from Google’s PageRank algorithm to models of DNA evolution. But though the solutions to his equation, called Markov numbers, are not nearly as well known, they arise in a vast range of mathematical disciplines, including combinatorics, number theory, geometry and graph theory.“It’s not just an equation, it’s a kind of method,” said Oleg Karpenkov, a mathematician at the University of Liverpool. “These numbers are central, deep inside mathematics … structures like this one are the kinds of ideas that are rare.”
The equation is this:$$x^2+y^2+z^2=3xyz$$Setting \(x=1\) and looking for suitable values of \(y\) and \(z\) yields the following triples (permalink): (1, 1, 1), (1, 1, 2), (1, 2, 5), (1, 5, 13), (1, 13, 34), (1, 34, 89), (1, 89, 233) and (1, 233, 610). All of these numbers are Fibonacci numbers. Specifically alternate Fibonacci numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610
Similarly, Setting \(x=2\) and looking for suitable values of \(y\) and \(z\) yields the following triples (permalink): (2, 5, 29), (2, 29, 169) and (2, 169, 985). These are alternating Pell numbers (apart from the initial adjacent 2 and 5. Specifically:
1, 2, 5, 12, 29, 70, 169, 408, 985
Pell numbers constitute OEIS A000129:
A000129 | Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2). |
The OEIS comments reveal a connection between these Markov numbers and the rational approximations to irrational numbers:
Also denominators of continued fraction convergents to \( \sqrt{2} \): 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, ...
The Quanta Magazine article continues:
It turns out that all the integer solutions to the equation are connected by a simple rule. Start with a solution \((a, b, c) \). Then the related triplet \((a, b, 3ab − c)\) is also a solution. The first two numbers stay the same, while \(c\), the third, is replaced by \(3ab − c\). Apply this rule to (1, 1, 1) and you get (1, 1, 2). Apply the rule again, and you’ll be back where you started, since 3 − 2 = 1. But if you flip the order of the numbers in the triple before applying the rule, it creates a whole universe of solutions. Input (1, 2, 1) and you’ll get (1, 2, 5).
Up until now, because of the identical 1's, the tree (shown in the illustration at the beginning of this story) doesn’t branch — the first few steps grow the trunk of the tree, so to speak. But if you start with a solution with three different numbers, like (1, 2, 5), the branches start to proliferate. Input (5, 1, 2) and you get (2, 5, 29). But (2, 5, 1) results in (1, 5, 13). (If you input (1,2,5) then the rule takes you back down to a lower branch of the tree.) From this point on, every solution has three different numbers, so every branch of the tree leads to two new branches.
The leftmost branch of the tree might look familiar — it contains every other number in the Fibonacci sequence, one of the best known in mathematics (each number in this sequence is the sum of the two previous terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, …). The rightmost branch similarly contains every other term in the Pell sequence, a related, if slightly less famous, sequence. The way these sequences appear in the tree of solutions is “one of the most beautiful things in mathematics I know,” said Alexander Gamburd, a professor at the City University of New York.
At this point, a diagram taken from this source is helpful. See Figure 2.
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Figure 2 |
The comments to the diagram are as follows:
(1,1,1) and (1,1,2) satisfy the Markov equation; all other solutions can be cataloged in a tree as shown below. The red lines partition the plane into open-ended regions, and each region is labeled with a number. The numbers in the three regions around any vertex constitute a triple \( (a,b,c) \) that satisfies the Markov equation. For example, the solutions (1, 2, 5), (2, 169, 985), and (34, 1325, 135137) are associated with vertices in the tree shown.
Simple algebra will show that if the triple \((a,b,c) \) satisfies the Markov equation, so does \( (a,b,3ab-c) \). This relationship is the basis for the construction of the tree; the four Markov numbers in the regions around any two connected vertices are related to one another as shown below. The angles of the red lines vary, and aren't important; what counts is the way the lines partition the plane into (open-ended) regions, and how each region is associated with a Markov number, as shown in Figures 3 and 4.
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Figure 3 |
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Figure 4 |
The article continues:
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 Markov equation \(a^2 + b^2 + c^2 = 3abc \).For example, 7561 = (1942 + 132) / 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.Markov proved that this form of tree will generate all possible Markov 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 Markov equation. This is known as the unicity (uniqueness) conjecture for the Markov numbers. Baragar and Button have published proofs of uniqueness for certain specific subsets of the Markov 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.Markov 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 Markov numbers, with an emphasis on material that doesn't require advanced math knowledge to appreciate.
Here is a link to a brief biography of Markov.
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Andrei Andreyevich Markov 1856 - 1922 |
References:
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