Approximating Markov’s equation
Markov numbers are integer solutions to
x^2 +y^2 +z^2 = 3xyz.
The Wikipedia article on Markov numbers mentions that Don Zagier studied Markov numbers by looking the approximating equation
x^2 +y^2 +z^2 = 3xyz+ 4/9
which is equivalent to
f(x) +f(y) = f(z)
wheref(t) is defined as arccosh(3t/2). It wasn't clear to me why the two previous equations are equivalent, so I'm writing this post to show that they are equivalent.
ExamplesBefore showing the equivalence of Zagier's two equations, let's look at an example that shows solutions to his second equation approximate solutions to Markov's equation.
The following code verifies that (5, 13, 194) is a solution to Markov's equation.
x, y, z = 5, 13, 194assert(x**2 + y**2 + z**2 == 3*x*y*z)
With the samex andy above, let's show that thez in Zagier's second equation is close to thez above.
from math import cosh, acoshf = lambda t: acosh(3*t/2)g = lambda t: cosh(t)*2/3z = g(f(x) + f(y))print(z)
This gives z = 194.0023, which is close to the value of z in the Markov triple above.
Applying Osborn's ruleNow suppose
f(x) +f(y) = f(z)
which expands to
arccosh(3x/2) + arccosh(3y/2) = arccosh(3z/2).
It seems sensible to apply cosh to both sides. Is there some identity for cosh of a sum? Maybe you recall the equation for cosine of a sum:
cos(a +b) = cos(a) cos(b) - sin(a) sin(b).
Then Osborn's rule says the corresponding hyperbolic identity is
cosh(a +b) = cosh(a) cosh(b) - sinh(a) sinh(b).
Osborn's rule also says that the analog of the familiar identity
sin^2(a) + cos^2(b) = 1
is
sinh^2(a) = cosh^2(b) - 1.
From these two hyperbolic identities we can show that [1]
cosh( arccosh(a) + arccosh(b) ) =ab + (a^2 - 1) (b^2 - 1).
Slug it outThe identity derived above is the tool we need to reduce our task to routine algebra.
If
arccosh(3x/2) + arccosh(3y/2) = arccosh(3z/2)
then
(3x/2) (3y/2) + ((3x/2)^2 - 1) ((3y/2)^2 - 1) = 3z / 2
which simplifies to Zagier's equation
x^2 +y^2 +z^2 = 3xyz + 4/9.
Related posts[1] The equation holds at least for x > 1 andy > 1, which is enough for this post. More general arguments run into complications due to branch cuts.
The post Approximating Markov's equation first appeared on John D. Cook.