Collatz, Erdős, and Tao

The Prompt

Explain the Collatz conjecture to someone who knows basic arithmetic but no advanced mathematics. State the rule; walk through the sequence for 27 (a famously long trajectory); cite Collatz's 1937 formulation, Erdős's remark about the readiness of mathematics, and Tao's 2019 partial result on almost all orbits. Explain what Tao proved and why the full conjecture remains open.

Completion Criteria

A short essay (three or four paragraphs) with at least three citations — Collatz (1937), Erdős (via Lagarias, 1985), and Tao (2019/2022) — and a worked example of the sequence for 27.

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