Assignment # 8 – Number Theory

(1) Find the simple continued fraction expansion of: 290/176 .
(2) Find the simple continued fraction expansion of square root of 11
(3) Show that [4, 2] = 2 + square root of 6
(4) Find [1, 2, 3].
(5) Show that if p > q and p/q = [a0; a1, … , an], then q/p = [0; a0; a1, … ,an].
(6) Find the convergents of [1, 3, 6, 3].
(7) Let alpha = [a0, a1, a2, …] be a positive irrational number.
(a) Show that a0 >= 0.
(b) Prove that an > 0 for all n >= 1.
(8) Show that the recurrence relations of Proposition 11.2.3 are valid for k = 0; 1 by
taking p-1 = 1, q-1 = 0, p-2 = 0, and q-2 = 1. Show that

assignment 8