The Simplest Math Problem No One Can Solve - Collatz Conjecture

Summary by Krater

An exploration of the Collatz conjecture (3n+1 problem), examining its origin, historical context, attempts at proof, and applications like Benford's Law and FRACTRAN.

From the video

Answers: What is the Collatz conjecture and why has it not been solved?

• Collatz conjecture
• 3n+1 problem
• Hailstone numbers
• Benford's Law
• Pólya conjecture
• FRACTRAN
• Directed graphs
• Turing machines
• Halting problem

What it concludes

• Every positive integer, when applying the 3x+1 and x/2 rules, will eventually end up in the 4-2-1 loop.
• The paths that different numbers take are randomness, exhibiting geometric Brownian motion.
• For the first billion sequences, 1 is by far the most common leading digit, obeying Benford's Law.
• Statistically speaking, 3x+1 sequences are more likely to shrink than grow because the geometric mean of growth steps is 3/4.
• No counterexample to the Collatz conjecture has been found by brute force up to 2 to the 68.
• Any loop other than 4-2-1 must be at least 186 billion numbers long.