A Markov number or Markoff number is a positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''x'', ''y'' or ''z'' that is part of a solution to the Markov
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...
:
studied by .
The first few Markov numbers are
:
1,
2,
5,
13,
29,
34,
89,
169
Year 169 ( CLXIX) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Senecio and Apollinaris (or, less frequently, year 922 ''Ab urbe co ...
,
194,
233, 433, 610, 985, 1325, ...
appearing as coordinates of the Markov triples
:(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), (1, 233, 610), (2, 169, 985), (13, 34, 1325), ...
There are infinitely many Markov numbers and Markov triples.
Markov tree
There are two simple ways to obtain a new Markov triple from an old one (''x'', ''y'', ''z''). First, one may
permute the 3 numbers ''x'',''y'',''z'', so in particular one can normalize the triples so that ''x'' ≤ ''y'' ≤ ''z''. Second, if (''x'', ''y'', ''z'') is a Markov triple then by
Vieta jumping so is (''x'', ''y'', 3''xy'' − ''z''). Applying this operation twice returns the same triple one started with. Joining each normalized Markov triple to the 1, 2, or 3 normalized triples one can obtain from this gives a graph starting from (1,1,1) as in the diagram. This graph is
connected; in other words every Markov triple can be connected to by a sequence of these operations. If we start, as an example, with we get its three
neighbors , and in the Markov tree if ''z'' is set to 1, 5 and 13, respectively. For instance, starting with 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.
All the Markov numbers on the regions adjacent to 2's region are
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
-indexed Pell numbers (or numbers ''n'' such that 2''n''
2 − 1 is a
square, ), and all the Markov numbers on the regions adjacent to 1's region are odd-indexed Fibonacci numbers (). Thus, there are infinitely many Markov triples of the form
:
where ''F''
''k'' is the ''k''th
Fibonacci number. Likewise, there are infinitely many Markov triples of the form
:
where ''P''
''k'' is the ''k''th
Pell number.
Proof that this generates all possible triples
Start with some solution (''x'', ''y'', ''z''), and assume all three are distinct. Now consider the
quadratic
:
Note that ''z'' is a
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
. By
Vieta's formulas
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta").
Basic formula ...
, the other root ''z′'' satisfies ''z + z′'' = 3''xy'' and ''zz′ = x''
 2 + ''y''
 2. Thus since ''z'' is positive, ''z′'' is also positive, we see that ''z′'' = 3''xy − z'' gives another solution.
Now,
WLOG, assume ''x'' > ''y'', then take
:
Since ''y'' > 0, 2 − 3''y'' ≤ −1, so ''f''(''x'') < 0. Since ''f''(''t'') is an upward-facing
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...
, that means min(''z'', ''z′'' ) < ''x'' < max(''z'', ''z′'' ).
That means that we can construct three new solutions: (''x'', ''y'', 3''xy − z''), (''x'', 3''xz − y'', ''z''), and (3''yz'' − ''x'', ''y'', ''z'') and these are ''distinct''. By our calculation above, exactly one of the three new solutions will have a smaller maximum element than (''x'', ''y'', ''z'') (and the other two larger).
Thus we proceed in this way, reducing the maximum element each time (this is
Vieta jumping). Since we are working with only positive integers, we must eventually stop, which means we reach a solution that has not all elements distinct.
It's left for us to consider such a solution. WLOG assume ''x'' = ''y'', then 2''x''
2 + ''z''
2 = 3''x''
2''z''. Thus ''x''
2 , ''z''
2 and ''x'' , ''z'', so write ''z'' = ''ax''. So we get
:
So we see ''a, 2'' so ''a'' = 1 or 2. If ''a'' = 1 then we get (1, 1, 1) and if ''a'' = 2 then we get (1, 1, 2). And from (1, 1, 2) we get to (1, 1, 1) by taking (''x'', ''y'', 3''xy − z'').
Thus we see that starting from an arbitrary solution we eventually come to (1, 1, 1), and so these are all the solutions.
Other properties
Aside from the two smallest ''singular'' triples (1, 1, 1) and (1, 1, 2), every Markov triple consists of three distinct integers.
The ''unicity conjecture'' states that for a given Markov number ''c'', there is exactly one normalized solution having ''c'' as its largest element:
proofs
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a co ...
of this
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in ...
have been claimed but none seems to be correct.
Odd Markov numbers are 1 more than multiples of 4, while
even
Even may refer to:
General
* Even (given name), a Norwegian male personal name
* Even (surname)
* Even (people), an ethnic group from Siberia and Russian Far East
**Even language, a language spoken by the Evens
* Odd and Even, a solitaire game wh ...
Markov numbers are 2 more than multiples of 32.
In his 1982 paper,
Don Zagier conjectured that the ''n''th Markov number is asymptotically given by
:
The error
is plotted below.
Moreover, he pointed out that
, an approximation of the original Diophantine equation, is equivalent to
with ''f''(''t'') =
arcosh
In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions.
For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. The ...
(3''t''  / 2). The conjecture was proved by
Greg McShane and
Igor Rivin __NOTOC__
Igor Rivin (born 1961 in Moscow, USSR) is a Russian-Canadian mathematician,
working in various fields of pure and applied mathematics, computer science,
and materials science. He was the Regius Professor of Mathematics at the Univers ...
in 1995 using techniques from
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
.
The ''n''th
Lagrange number can be calculated from the ''n''th Markov number with the formula
:
The Markov numbers are sums of (non-unique) pairs of squares.
Markov's theorem
showed that if
:
is an
indefinite binary quadratic form with
real coefficients and
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the orig ...
, then there are integers ''x'', ''y'' for which ''f'' takes a nonzero value of
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
at most
:
unless ''f'' is a ''Markov form'': a constant times a form
:
such that
: