Looping and Divergence in the Collatz Conjecture

In this paper, we investigate the possible scenarios in which a number does not satisfy the Collatz Conjecture. Specifically, we examine numbers which may have a looping Collatz reduction sequence as well as numbers which may lead to a diverging Collatz reduction sequence. In order to investigate these, we look at the parity of the numbers in a general Collatz reduction sequence. Further, we examine cases in which these parity cycles repeat themselves infinitely in the reduction sequence. Through the research conducted in the paper, we formulate a necessary condition for looping in the Collatz Conjecture. We also prove that if a number has a diverging reduction sequence, then it must generate an infinite non-repeating parity cycle.