Mathematics July 22, 2026 · 10 min read

The Collatz Conjecture: Mathematics' Most Famous Unsolved Problem

Explore the Collatz conjecture (3n+1 problem). Learn the rules, see worked examples, and understand why this simple problem remains unsolved.

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The Collatz conjecture is one of mathematics' most tantalizing unsolved problems. Despite its simple statement — take any positive integer, and repeatedly apply one of two easy rules — no one has been able to prove that the process always reaches 1.

The Rules

Start with any positive integer n:

  • If n is even, divide it by 2: n — n/2
  • If n is odd, multiply by 3 and add 1: n — 3n + 1
  • Repeat until you reach 1

Example: Starting from 6

  1. 6 — 3 (even: divide by 2)
  2. 3 — 10 (odd: 3×3+1)
  3. 10 — 5 (even: divide by 2)
  4. 5 — 16 (odd: 3×5+1)
  5. 16 — 8 — 4 — 2 — 1
  6. Sequence complete in 8 steps

Why Is It Unsolved?

The problem is deceptively simple to state but incredibly difficult to prove. The sequence can spike to very large values before eventually descending. For example, starting at 27, the sequence reaches a peak of 9,232 before eventually reaching 1 after 111 steps.

Computational Verification

The conjecture has been verified computationally for all starting values up to 2^68 (approximately 2.95 × 10²⁰). Despite this extensive testing, no general proof exists.

Connections to Other Fields

  • Dynamical systems: The Collatz function is a discrete dynamical system
  • Computational complexity: Generalizing the problem is Turing-complete
  • Number theory: Deep connections to modular arithmetic