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Encyclopedia > Markov number
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A Markov number or Markoff number is an integer x, y or z that is part of a solution to the Markov Diophantine equation In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ...

x2 + y2 + z2 = 3xyz

The first few Markov numbers are


1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325, ... (sequence A002559 in OEIS) Jump to: navigation, search Look up one on Wiktionary, the free dictionary. ... 2 (two) is a number, numeral, and glyph. ... 5 (five) is a number, numeral, and glyph. ... Jump to: navigation, search 13 (Thirteen) is the natural number following 12 and preceding 14. ... Jump to: navigation, search 29 (twenty-nine) is the natural number following 28 and preceding 30. ... 34 is the natural number following 33 and preceding 35. ... 89 (eighty-nine) is the natural number following 88 and preceding 90. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...


appearing in the solutions


(1, 1, 1), (1, 1, 2), (1, 2, 5), (1, 5, 13), (2, 5, 29), (1, 13, 34), (1, 34, 89), (2, 29, 169), (5, 13, 194), (1, 89, 233), (5, 29, 433), (89, 233, 610), etc.

The Markov numbers, arranged in a binary tree.
Enlarge
The Markov numbers, arranged in a binary tree.

There are infinitely many Markov numbers and Markov triples. Any Markov number appears in at least three solutions, but is the largest integer in only one solution. (A030452 lists Markov numbers that appear in solutions where one of the other two terms is 5).


Due to the commutative properties of addition and multiplication, the solutions may be arranged in any order, but it might be helpful to arrange each Markov triple in ascending order, and the triples in order by highest integer contained (as above).


The Markov numbers can also be arranged in a binary tree. The largest number at any level is always about a third from the bottom. All the Markov numbers on the regions adjacent to 2's region are odd-indexed Pell numbers (or numbers n such that 2n2 − 1 is a square, A001653), and all the Markov numbers on the regions adjactent to 1's region are odd-indexed Fibonacci numbers (A001519). Thus, there are infinitely many Markov triples of the form (1,F2n − 1,F2n + 1), where Fx is the xth Fibonacci number. Likewise, there are infinitely many Markovo tripli of the form (1,P2n − 1,P2n + 1), where Px is the xth Pell number. In mathematics, the Pell numbers and Companion Pell Numbers (Pell-Lucas Numbers) are both sequences of integers. ... Jump to: navigation, search In mathematics, the Fibonacci numbers form a sequence defined recursively by: In other words: one starts with 0 and 1, and then produces the next Fibonacci number by adding the two previous Fibonacci numbers. ...


Knowing one Markov triple (x, y, z) one can find another Markov triple, of the form (x,y,3xyz). Markov numbers are not always prime, but members of a Markov triple are always coprime (with the exception of the first two triples). It's not necessary that x < y < z in order for the (x,y,3xyz) to yield another triple. In fact, if one doesn't change the order of the members before applying the transform again, it returns the same triple one started with. Thus, starting with (1, 1, 2) and trading y and z before each iteration of the transform lists Markov triples with Fibonacci numbers. Starting with that same triplet and trading x and z before each iteration gives the triples with Pell numbers. Coprime - Wikipedia /**/ @import /skins-1. ...


Markov numbers are named after the Russian mathematician Andrey Markov. Due to the different but equally valid ways of transliterating Cyrillic, the term is written as "Markoff numbers" in some literature. But in this particular case, "Markov" might be preferrable because "Markoff number" might be misunderstood as "mark-off number." This is an article about Russian mathematician Andrey Markov. ...


 
 

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