Talk:Markov number

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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)

[edit] 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)

[edit] 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)

[edit] 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)