# diophantine equation in positive integers

solve this equation in positive integers:

$x^2+y^2+z^2=3xyz$

I could prove that it's solutions are infinite, for if $(x,y,z)$ is a solution, with $x\le y \le z$, then $(3yz-x,y,z)$ is also a solution with $\max(x,y,z)< \max(3yz-x,y,z)$ and also $(1,1,1)$ is a solution, so we can build an infinite set of solutions, but I'm stuck to show that these are the only solutions. I'll be glad if you help me.

• This is the Markov equation, isn't it? Much should be available by searching for that term (possibly under the alternate spelling, Markoff). Commented Dec 27, 2011 at 15:17
• See M.G. Krein, "Markov's Diophantine Equation", pp. 121-126 in Kvant Selecta: Algebra and Analysis I (S. Tabachnikov ed.), Amer. Math. Soc., Providence, 1991.
– KCd
Commented Dec 27, 2011 at 16:32

Edit, April 2015: the solutions of the diophantine equation are discussed at http://en.wikipedia.org/wiki/Markov_number while the spectrum and the book by Cusick and Flahive are mentioned at http://en.wikipedia.org/wiki/Markov_spectrum

ORIGINAL: This is a really nice topic. The punchline is that one finds all binary quadratic forms, integer coefficients $f(x,y) = a x^2 + b x y + c y^2$ and indefinite, so that the discriminant $\Delta = b^2 - 4 a c$ is positive but not allowed to be a square, such that the "Markov Ratio" is below 9.

The trick is that every such form $f(x,y)$ can only represent $0$ when both $x=0, y=0.$ So, for at least one of the variables nonzero, the form takes a nonzero value, maybe positive maybe negative. Either way, there is a positive minimum of the form, call it $M_f,$ such that $|f(x,y)| \geq M_f$ whenever $x,y$ are integers and not both 0.

What Kaplansky and I called the "Markov Ratio," terminology that is destined to die out, was $$\frac{\Delta}{M_f^2}$$ can only be smaller than 9 for very special forms. I will report the terminology in Cusick and Flahive, The Markoff and Lagrange Spectra. Let us have the Markov equation $$m^2 + m_1^2 + m_2^2 = 3 m m_1 m_2$$ in positive integers. For each such $m,$ define a positive integer $u$ to be the least positive solution to $$\pm m_2 x \equiv m_1 \pmod m.$$ Next, define $v$ by $u^2 + 1 = v m.$ The Markov forms are given by $$f(x,y) = m x^2 + (3m-2u)xy + (v-3u)y^2$$ and give an integral equivalence class of every form for which $$\frac{\Delta}{M_f^2} < 9.$$ The main theorem is that $$M_f = m.$$ A short and understandable proof is in Cusick and Flahive, pages 20-22, Theorem 2, especially part (C).

Checking, we find $\Delta = 9 m^2 - 4.$ So, indeed $$\frac{\Delta}{M_f^2} = 9 - \frac{4}{m^2}.$$ This is intimately connected with continued fractions, of course.

I am especially proud of finding a set of mirror forms, $$g(x,y) = m^2 x^2 + m(3m-2u)xy + (u^2-3um -1)y^2$$ with $$M_f = m^2, \; \; \Delta = 9 m^4 + 4 m^2, \; \; \frac{\Delta}{M_f^2} = 9 + \frac{4}{m^2}.$$ These were mentioned briefly as Section 8.3 of Indefinite Binary Quadratic Forms With Markov Ratio Exceeding 9 by me and Kap, Illinois Journal of Mathematics, volume 47, 2003, pages 305-316. The proof this time was not included but is similar to that section in Cusick and Flahive.

Another quirky note: all the Markov discriminants $9 m^2 - 4$ can be written as the sum of two squares. You don't see that every day.

ADDENDUM: the quadratic forms I have displayed are already "reduced," which means the continued fraction for the larger root $x/y$ is purely periodic, see How to detect when continued fractions period terminates

• I'm very interested to see this material. I wonder, since you have studied the Markov (or Markoff) spectrum , if you have also studied the geometric results starting with Harvey Cohn and continued with Caroline Series , etc . I'm leaving a few people out I'm sure.I studied this material for a senor thesis with Marvin Greenberg at UCSC many many years ago. Later in graduate school I bugged Troels Jorgenson to tell me his trace condition for geodesics to have self intersection. I was generating pictures of these geodesics which have self intersections on the once punctured torus . out of text.
– Alan
Commented Sep 30, 2013 at 22:45
• @Alan, the Markov paper is a pdf at zakuski.utsa.edu/~jagy/bib.html Other than my answers on MSE and MO, the best source on indefinite binary forms is Duncan Buell, Binary Quadratic Forms. That is where I found out about the advantages of computing with reduced forms...Cusick and Flahive come from a different direction, I haven't really seen that material anywhere else. Commented Sep 30, 2013 at 22:51
• @Will, where do we get $M_f$ = m for a markoff form f? I have been trying to figure this part out for awhile now. Commented Apr 15, 2015 at 23:10
• @user70363, if you are asking about how to prove that $m$ really is the "minimum," that is in the book by Cusick and Flahive, The Markoff and Lagrange Spectra. The proof is elegant. Commented Apr 15, 2015 at 23:30
However, there are other solutions not in the above series. If (x, y, z) is a solution with $x \le y \le z$, then (3xz – y, x, z) is also a solution. (3xz – y) must be positive since from $x^2 + y^2 + z^2 = 3xyz$ it follows that $y^2 < (3xz)y$ and therefore (given that y is positive) y < 3xz.