Minutes 12/04

Notetaker for Tuesday: Martin
Notetaker for Thursday: Emilie
No Snacks on Tuesday

Stephen suggests that we should look at the rational parametrization of the 
Markoff equation.

We all go to the Computer Lab to work on various problems.

Emilie looked at the lattice that corresponds to the Markoff Brother, Groucho,
a^2+2b^2+3c^2=6abc.

Sam, Carl and Paul look at Newton's approximation for quadratic roots.

Break for Coke and Doughnuts. (Conversation about the philosophy of 
"discovering" versus "inventing" mathematical theorems.)

After working most of the time with Newton's method of approximation on 
quadratic equations, Sam, Paul and Carl come up with a numerically supported 
conjecture for the coefficients of the numerator and the denominator of the 
nth approximation.

Let n be the number of times we iterate Newton's approximation of ax^2+bx+c.  
Then the coefficient of x^k in both the numerator and the denominator is 
binomial(2^n , k) (times some polynomial in a,b,c).

Stephen found a rational parametrization of the markoff equation in P3.  
The parametrization yields the triple 
     ( (a^2+b^2+c^2)/3bc, (a^2+b^2+c^2)/3ac, (a^2+b^2+c^2)/3ab ).  
For a given a, b, c, these numbers satisfy the Markoff equation, but are 
not necessarily integers.