Number theory · Maths EE idea · Solid
Patterns in Collatz stopping times
A research question to start from
What can be proved about the stopping times of the Collatz map for numbers in particular residue classes, even though the conjecture itself is open?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
An open problem where you can still prove real statements, provided you are honest about what remains conjecture.
Mathematics you would need
- Iterated functions
- Residue classes modulo powers of 2
- Proof by induction
- Data analysis of computed sequences
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Compute stopping times up to a large bound and look for structure by residue class.
- Prove results such as: numbers in certain residue classes modulo 2^k (for example n ≡ 1 mod 4) must fall below their starting value within a fixed number of steps.
- Discuss heuristics for why most numbers decrease and where they fall short of a proof.
Scope and difficulty
Solid. Risky if you aim at the conjecture; good if the question targets provable partial results.
Pitfalls
- Claiming or implying a proof of the conjecture.
- Pages of data with no analysis.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Collatz conjecture stopping time residue classes; 3x+1 heuristic. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).