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.