Exercises

5..

1. If $c_n=p_n/q_n$ is the $n$ th convergent of $[a_0,a_1,\ldots,a_n]$ and $a_0>0$ , show that
   $$[a_n,a_{n-1},\ldots, a_1, a_0] = \frac{p_n}{p_{n-1}}$$
   and
   $$[a_n,a_{n-1},\ldots, a_2, a_1] = \frac{q_n}{q_{n-1}}.$$
   (Hint: In the first case, notice that $$\frac{p_n}{p_{n-1}} = a_n + \frac{p_{n-2}}{p_{n-1}} = a_n + \frac{1}{\frac{p_{n-1}}{p_{n-2}}}.$$ )

2. Show that every nonzero rational number can be represented in exactly two ways be a finite simple continued fraction. (For example, $2$ can be represented by $[1,1]$ and $[2]$ , and $1/3$ by $[0,3]$ and $[0,2,1]$ .)

3. Evaluate the infinite continued fraction $[2,\overline{1,2,1}]$ .

4. Determine the infinite continued fraction of $\frac{1+\sqrt{13}}{2}$ .

5. Let $a_0\in\mathbb{R}$ and $a_1,\ldots,a_n$ and $b$ be positive real numbers. Prove that
   $$[a_0,a_1,\ldots,a_n+b] < [a_0,a_1,\ldots,a_n]$$
   if and only if $n$ is odd.

6. (*) Extend the method presented in the text to show that the continued fraction expansion of $e^{1/k}$ is
   $$[1, (k-1), 1, 1, (3k-1), 1, 1, (5k-1), 1, 1, (7k-1),\ldots]$$
   for all $k \in \mathbb{N}$ .
   1. Compute $p_0$ , $p_3$ , $q_0$ , and $q_3$ for the above continued fraction. Your answers should be in terms of $k$ .

   2. Condense three steps of the recurrence for the numerators and denominators of the above continued fraction. That is, produce a simple recurrence for $r_{3n}$ in terms of $r_{3n-3}$ and $r_{3n-6}$ whose coefficients are polynomials in $n$ and $k$ .

   3. Define a sequence of real numbers by
      $$T_n(k)=\frac{1}{k^n}\int_{0}^{1/k}\frac{(kt)^{n}(kt-1)^{n}}{n!}\phantom{1} e^tdt.$$
      1. Compute $T_0(k)$ , and verify that it equals $q_0e^{1/k}-p_0$ .
      2. Compute $T_1(k)$ , and verify that it equals $q_3e^{1/k}-p_3$ .
      3. Integrate $T_n(k)$ by parts twice in succession, as in Section 5.3, and verify that $T_n(k)$ , $T_{n-1}(k)$ , and $T_{n-2}(k)$ satisfy the recurrence produced in part 6b, for $n\geq 2$ .

   4. Conclude that the continued fraction
      $$[1, (k-1), 1, 1, (3k-1), 1, 1, (5k-1), 1, 1, (7k-1),\ldots]$$
      represents $e^{1/k}$ .

7. Let $d$ be an integer that is coprime to $10$ . Prove that the decimal expansion of $\frac{1}{d}$ has period equal to the order of $10$ modulo $d$ . (Hint: For every positive integer $r$ , we have $\frac{1}{1-10^r} = \sum_{n\geq 1} 10^{-rn}.$ )

8. Find a positive integer that has at least three different representations as the sum of two squares, disregarding signs and the order of the summands.

9. Show that if a natural number $n$ is the sum of two two rational squares it is also the sum of two integer squares.

10. (*) Let $p$ be an odd prime. Show that $p\equiv 1,3\pmod{8}$ if and only if $p$ can be written as $p=x^2 + 2y^2$ for some choice of integers $x$ and $y$ .

11. Prove that of any four consecutive integers, at least one is not representable as a sum of two squares.