Homework Assignment 5 Due Wednesday October 30

William Stein

Math 124  Harvard University  Fall 2002

Instructions: Please work with others. There are six problems.

1. (4 points) Construct a sequence of positive real numbers a_{0}, a_{1}, a_{2}, … such that

lim_{n→∞} c_{2n} ≠ lim_{n→∞} c_{2n+1},

where c_{n} = [a_{0}, …, a_{n}] is the nth partial convergent. To get full credit on this problem, you do not have to compute the limits for your example, though it would be cool if you were to do so.

2. (2 points) Find two distinct continued fractions a_{0}, a_{1}, a_{2}, … and b_{0}, b_{1}, b_{2}, … such that

[a_{0}, a_{1}, a_{2}, …] = [b_{0}, b_{1}, b_{2}, …].

(Note that necessarily the a_{i} and b_{i} won’t all be integers.)

3. (a) (3 points) Find the continued fraction expansion of (1 + 2\sqrt{3})/4. Prove that your answer is correct.

(b) (3 points) Evaluate the infinite continued fraction [0, \overline{1, 3}]

4. (5 points) Let a_{0} ∈ ℝ and a_{1}, …, a_{n} and b be positive real numbers. Prove that

[a_{0}, a_{1}, …, a_{n} + b] < [a_{0}, a_{1}, …, a_{n}]

if and only if n is odd.

5. Let s(n) = 1 + 2 + ⋯ + n = \frac{n(n+1)}{2} be the sum of the first n positive integers.

(a) (2 points) Use MAGMA to find all n < 10000 such that s(n) is a perfect square.

(b) (5 points) Prove that there are infinitely many n such that s(n) is a perfect square. (Hint: Find a relationship between such n and solutions to a certain Pell’s equation.)

6. Let α be a real number, and let p_{k}/q_{k} denote the partial convergents of the integral continued fraction for α.

(a) (2 points) Prove that for every k ≥ 0, |α − \frac{p_{k}}{q_{k}} | < q_{k}^{2}.

(b) (5 points) Let the decimal expansion of α be

α = b + \frac{b_{1}}{10} + \frac{b_{2}}{10^{2}} + \frac{b_{3}}{10^{3}} + \frac{b_{4}}{10^{4}} + ⋯,

where 0 ≤ b_{n} ≤ 9 for all n. Suppose that for some convergent p_{k}/q_{k} we have q_{k} = 100. Prove that either b_{3} = b_{4} = 0 or b_{3} = b_{4} = 9. (This problem is from page 210 of Stark’s book An Introduction To Number Theory.)