Talk:Markov number

Brackets

To delimit Markov triples, Mathworld uses parentheses

(1, 1, 1), (1, 1, 2), ...

though a Mathematica command would most likely return

{{1, 1, 1}, {1, 1, 2}, ...

and I'm guessing that's why PrimeFan chose to do it that way. But it doesn't seem quite right. In the math tags, you have to "escape" any curly brackets you want to show.

So what's the correct way of delimiting Markov triples? With parentheses or with curly brackets? Anton Mravcek 21:57, 25 October 2005 (UTC)

Usually, curly brackets denote sets and parentheses denote tuples. In a set, every entry can occur only once, so I'd say it should be parentheses. This is also what [1] uses. -- Jitse Niesen (talk) 22:21, 25 October 2005 (UTC)

I've gone ahead and changed the brackets to parentheses. Anton Mravcek 20:38, 27 October 2005 (UTC)

It's precisely because of Mathematica that I used curly brackets. I simply copied and pasted the Markov triples. I didn't give it a second thought. Perhaps per analogy to Brown numbers we should use parentheses here. PrimeFan 17:25, 27 October 2005 (UTC)

Markov primes

I removed the little bit about Markov primes. Maybe there is some interesting relation between a Markov number that is a prime and its index in the Markov sequence, or some other interesting property of such numbers. But I don't think there are any professional mathematicians researching Markov primes, nor are any large prime discoverers making an effort to identify large Markov primes. Anton Mravcek 20:38, 27 October 2005 (UTC)

Interesting Preprint

Ying Zhang, Congruence and Uniqueness of Certain Markoff Numbers, 2006.

Abstract: By making use of only simple facts about congruence, we first show that every even Markoff number is congruent to 2 modulo 32, and then, generalizing an earlier result of Baragar, establish the uniqueness for those Markoff numbers c where one of 3c - 2 and 3c + 2 is a prime power, 4 times a prime power, or 8 times a prime power.

That is a very interesting preprint. Thanks for bringing it to our attention. PrimeFan 23:49, 20 July 2007 (UTC)

I also wish to extend thanks to Mnp. I've added the facts about the congruences by parity to the article. I have not yet added the stuff about uniqueness. Anton Mravcek 19:51, 22 July 2007 (UTC)

How to solve it???

Can anyone of you tell me how to solve the equation x^2 + y^2 + z^2 = 3xyz 125.234.150.44 08:51, 20 July 2007 (UTC)

Can you be a little more specific? Are you asking how to solve it if one knows one variable or two variables? Or are you asking for a way to find triples without knowledge of any specific Markov numbers?

If you know three of them, you can find another one just by plugging them into the equation (x, y, 3xy − z). PrimeFan 23:56, 20 July 2007 (UTC)

Value for C in asymptotic formula

The value C = 2.3523418721 quoted in the asymptotic formula for $m_{n}$ was computed from the value 0.18071704711507 for C^{-2} in Zagier's 1982 paper. Unfortunately that value is inaccurate, most importantly due to an accidentally missing digit, and in fact a more accurate value is used to compute other numbers in that paper. The value should be 0.180717104711806 for C^{-2} and hence C = 2.3523414972 (to 10 places). See OEIS: A261613 for more details. Chris Thompson (talk) 20:08, 20 April 2016 (UTC)

In the absence of objections, I have updated the main page accordingly. Chris Thompson (talk) 19:24, 23 April 2016 (UTC)

Status of the asymptotic formula

The main page states that the conjecture that $\textstyle m_{n}={\tfrac {1}{3}}e^{C{\sqrt {n}}+o(1)}$ was "proved by Greg McShane and Igor Rivin in 1995". I think this is incorrect. That conjecture is equivalent to

$M(x)=C(\log(3x))^{2}+o(\log x)$

(not the same C) where $M(x)$ is the number of Markov numbers less than $x$. Don Zagier's 1982 paper proved this with an error term of $\textstyle O(\log x(\log \log x)^{2})$. The McShane and Rivin paper referenced (which incidentally is most easily accessed as arXiv:math/0005220) improves this to $\textstyle O(\log x\log \log x)$, which is still some way from $\textstyle o(\log x)$. In fact such improved estimates of the error term are clearly described in the paper as conjectures, even prompting the authors to remark at the end of the French summary "Nous croyons que la deuxieme conjecture est tres difficile."

In fact the conjecture appears to be still open. The recent paper arXiv:1603.06267 by Gamburd, Magee and Ronan states (Theorem 1) that "The best current result is due to McShane and Rivin", repeating $\textstyle O(\log x\log \log x)$ and giving a reference to their other 1995 paper, accessible as arXiv:math/0005222. Chris Thompson (talk) 16:17, 22 April 2016 (UTC)

The last sentence of the paper is: "This is contrary to Zagier’s conjecture which was also stated for the modular torus." So shouldn't the article say the conjecture was DISproved by McShane and Rivin?147.142.164.121 (talk) 09:53, 7 August 2020 (UTC)

Is it error?

Sentence: the Markov tree if x is set to 1, 5 and 13, respectively

I think it must be if z is set Jumpow (talk) 19:32, 12 April 2017 (UTC)

I agree, and have made the change Chris Thompson (talk) 13:08, 3 May 2017 (UTC)

Prime factors of Markov numbers

It is stated in the OEIS page that the odd prime factors of Markov numbers are congruent to 1 modulo 4. Is this an obvious fact? Thanks! 14.52.231.91 (talk) 05:28, 23 August 2024 (UTC)

Let $p$ be an odd prime that divides the element $z$ of the Markov triple $(x,y,z)$. Reducing Markov's equation mod $p$ gives $x^{2}=-y^{2}{\pmod {p}}$. This implies that $-1$ is a square mod $p$ and hence that $p\equiv 1{\pmod {4}}$. Will Orrick (talk) 14:09, 23 August 2024 (UTC)

Thanks! So should this be mentioned in the Wikipedia page? As a substitution of "odd Markov numbers are 1 more than multiples of 4"? 14.52.231.91 (talk) 00:54, 26 August 2024 (UTC)

I agree that it should be added. A more general restriction arises as follows: regard Markov's equation as a quadratic equation in $y$ and complete the square to get $(2y-3xz)^{2}=(9z^{2}-4)x^{2}-4z^{2}$. Now examine this equation modulo a prime $p\equiv 3{\pmod {4}}$. If $z\equiv 0{\pmod {p}}$ we get the contradiction that $-4x^{2}$ is a square mod $p$—leading to the same conclusion as before that $z$ cannot be divisible by a prime congruent to 3 mod 4. If $9z^{2}-4\equiv 0{\pmod {p}}$ we get the contradiction that $-4z^{2}$ is a square mod $p$. Hence if $z$ is a Markov number then $9z^{2}-4$ cannot be divisible by a prime congruent to 3 mod 4. This also could be included. Will Orrick (talk) 01:20, 26 August 2024 (UTC)

That's a nice property. Thanks! 14.52.231.91 (talk) 04:03, 26 August 2024 (UTC)