Physical Sciences and Mathematics Commons^™

Parity Periodicity: An Eliminative Approach To The Collatz Conjecture, Austin J. Phillips

Honors Theses

The 3n + l Conjecture states that when the Collatz function is applied repeatedly to an initial value, the sequence of values generated always converges to 1, regardless of the starting value. This paper strengthens the claim that all such sequences are convergent by showing that certain types of nonconvergent sequences cannot exist. Specifically, no sequence with parity-periodic values can exist This eliminates all possible nontrivially periodic sequences and all divergent sequences with periodic parity. Therefore, if a counterexample to the conjecture exists, It must be a divergent sequence whose values display no parity periodicity.