  Proposed in 1937, the Collatz conjecture has remained in the spotlight for
mathematicians and computer scientists alike due to its simple proposal, yet
intractable proof. In this paper, we propose several novel theorems,
corollaries, and algorithms that explore relationships and properties between
the natural numbers, their peak values, and the conjecture. These contributions
primarily analyze the number of Collatz iterations it takes for a given integer
to reach 1 or a number less than itself, or the relationship between a starting
number and its peak value.
