Talk:Markov number

message: 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)
replies: text: 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  uses. -- Jitse Niesen (talk) 22:21, 25 October 2005 (UTC)
replies: text: I've gone ahead and changed the brackets to parentheses. Anton Mravcek 20:38, 27 October 2005 (UTC)

text: 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)
thread_title: Brackets

message: 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)
thread_title:  Markov primes 

message: 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)
replies: text: 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)
thread_title:  Interesting Preprint 

message: 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  for more details.
Chris Thompson (talk) 20:08, 20 April 2016 (UTC)
replies: text: In the absence of objections, I have updated the main page accordingly. Chris Thompson (talk) 19:24, 23 April 2016 (UTC)
thread_title:  Value for C in asymptotic formula 

message: The main page states that the conjecture that \textstyle m_n = \tfrac13 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
) 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  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 .
Chris Thompson (talk) 16:17, 22 April 2016 (UTC)
replies: text: 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)
thread_title:  Status of the asymptotic formula 

message: 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)
replies: text: I agree, and have made the change Chris Thompson (talk) 13:08, 3 May 2017 (UTC)
thread_title:  Is it error?