Starting with any given positive integer $a > 1$ the following game is played. If $a$ is a perfect square, take its square root. Otherwise take $a + 3$. Repeat the procedure with the new positive integer (i.e., with $\sqrt { a }$ or $a + 3$ depending on the case). The resulting set of numbers is called the trajectory of $a$. For example the set $\{ 3, 6, 9 \}$ is a trajectory: it is the trajectory of each of its members. Which numbers have a finite trajectory? Possible hint: Find the set $$\{ n \mid n \text{ is the smallest number in some trajectory } S \}.$$ If you wish, you can get partial credit by solving the following simpler questions. (a) Show that there is no trajectory of cardinality 1 or 2. (b) Show that $\{ 3, 6, 9 \}$ is the only trajectory of cardinality 3. (c) Show that for any integer $k \geq 3$, there is a trajectory of cardinality $k$. (d) Find an infinite trajectory.
Starting with any given positive integer $a > 1$ the following game is played. If $a$ is a perfect square, take its square root. Otherwise take $a + 3$. Repeat the procedure with the new positive integer (i.e., with $\sqrt { a }$ or $a + 3$ depending on the case). The resulting set of numbers is called the trajectory of $a$. For example the set $\{ 3, 6, 9 \}$ is a trajectory: it is the trajectory of each of its members.
Which numbers have a finite trajectory? Possible hint: Find the set
$$\{ n \mid n \text{ is the smallest number in some trajectory } S \}.$$
If you wish, you can get partial credit by solving the following simpler questions.\\
(a) Show that there is no trajectory of cardinality 1 or 2.\\
(b) Show that $\{ 3, 6, 9 \}$ is the only trajectory of cardinality 3.\\
(c) Show that for any integer $k \geq 3$, there is a trajectory of cardinality $k$.\\
(d) Find an infinite trajectory.