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.