Consider a positive integer a>1. If a is not a perfect square then at the next move we add 3 to it and if it is a perfect square we take the square root of it. Define the trajectory of a number a as the set obtained by performing this operation on a. For example the cardinality of 3 is {3,6,9}.
Find all n such that the cardinality of n is finite. The following part problems may attract partial credit.<spanclass=′latex−bold′>(a)</span>Show that the cardinality of the trajectory of a number cannot be 1 or 2.
<spanclass=′latex−bold′>(b)</span>Show that {3,6,9} is the only trajectory with cardinality 3.
<spanclass=′latex−bold′>(c)</span> Show that there for all k≥3, there exists a number such that the cardinality
of its trajectory is k.
<spanclass=′latex−bold′>(d)</span> Give an example of a number with cardinality of trajectory as infinity. functionnumber theorystrong inductionCMIIMO