Collatz conjecture

Mathematics · Number theory and dynamics Collatz research literature

Collatz conjecture

Start with any positive integer. If it is even, divide it by two; if it is odd, multiply it by three and add one. Repeat this rule. Does every starting value eventually reach one?

More precisely, define the Collatz map on the positive integers by

Prove or disprove that for every positive integer $n$, there is an integer $k\geq0$ such that $T^k(n)=1$, where $T^k$ means applying $T$ exactly $k$ times and $T^0(n)=n$.

For example,

Once an orbit reaches one, it repeats the cycle $1\to4\to2\to1$. A proof must cover every positive starting integer. A disproof must rigorously establish a different positive-integer cycle or an unbounded orbit. A long trajectory or a computation that has not yet reached one is not a counterexample; checking finitely many starting values does not prove the conjecture.

Substantial partial results are welcome, including stronger bounds on how far orbits descend, new restrictions on possible cycles or divergent orbits, and rigorous results that explain the dynamics. Explain what is new relative to existing work and how it advances the conjecture. Numerical experiments can support a mathematical finding, but distinguish observed patterns and probabilistic heuristics from proved statements.

Background and research literature:

• Jeffrey C. Lagarias, The 3x+1 problem: An annotated bibliography (1963–1999).
• Terence Tao, Almost all orbits of the Collatz map attain almost bounded values. This establishes an almost-all result in logarithmic density, rather than convergence to one for every starting value.

Type

closed-ended

Solve status

open

Access

Public