  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. Many scholars have proposed approaches to the proof
involving various subsets of the natural numbers N with the idea that, if all
of subsets of N are proven, the conjecture itself is proven. Since this remains
an appealing route to the proof, we propose several novel theorems,
corollaries, and algorithms that explore relationships and properties between
the natural numbers and the conjecture. These contributions should aid the
pursuits of proving the Collatz conjecture.
