The Simplest Math Problem No One Can Solve - Collatz Conjecture
Veritasium・18 minutes read
The Collatz conjecture involves applying rules to numbers to reach a four, two, one loop and has various names like the Ulam conjecture and Syracuse problem. Mathematicians have made progress on the problem, confirming that almost all sequences reach a point below their initial value but still lack a definitive proof.
Insights
- The Collatz conjecture, also known as the Ulam conjecture or Kakutani's problem, involves applying rules to numbers to reach a loop of four, two, and one, generating hailstone numbers that eventually all reach one after varying paths, with some exceeding Mount Everest's height.
- Extensive testing has been conducted up to two to the 68, showing that all numbers eventually reach one, but a definitive proof is still lacking, highlighting the complexity and potential undecidability of the Collatz conjecture despite progress in understanding its patterns and behaviors.
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Recent questions
What is the Collatz conjecture in mathematics?
The Collatz conjecture is an unsolved problem that involves applying rules to numbers to reach a loop of four, two, one. It is also known by other names like the Ulam conjecture, Kakutani's problem, and the Syracuse problem.
What are hailstone numbers in mathematics?
Hailstone numbers are the numbers generated by the 3x+1 rule in the Collatz conjecture. These numbers eventually reach one, with paths varying widely in their ascent and descent.
How do odd numbers behave in 3x+1 sequences?
Odd numbers in 3x+1 sequences tend to shrink rather than grow due to the rules applied. This behavior is statistically observed in the sequences.
What visualization methods are used for Collatz sequences?
Visualization methods like directed graphs and coral-like structures are used to represent the paths of numbers in Collatz sequences, aiding in understanding the patterns and behaviors of the sequences.
What is the significance of Terry Tao's proof in the Collatz conjecture?
Terry Tao's proof supports the conjecture that almost all numbers have a number in their sequence that is arbitrarily small. However, this does not definitively prove that all numbers follow this criteria, leaving the conjecture still unsolved.
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Summary
00:00
Unsolved Collatz Conjecture: Numbers' Loop Mystery
- The Collatz conjecture is a famous unsolved problem in mathematics, involving applying rules to numbers to reach the four, two, one loop.
- The problem has various names like the Ulam conjecture, Kakutani's problem, and the Syracuse problem.
- The numbers generated by the 3x+1 rule are called hailstone numbers, eventually reaching one.
- The paths hailstone numbers take vary widely, with some climbing to heights higher than Mount Everest before falling back to one.
- Mathematicians have struggled with the problem, with some considering it a distraction from real math.
- Benford's law, a distribution pattern, is observed in 3x+1 sequences and various other contexts.
- Odd numbers in 3x+1 sequences statistically tend to shrink rather than grow due to the rules applied.
- Visualization methods like directed graphs and coral-like structures help represent the paths of numbers in 3x+1 sequences.
- Extensive testing has been done on numbers up to two to the 68, confirming that they all eventually reach one.
- Mathematicians have made progress in showing that almost all Collatz sequences reach a point below their initial value, but a definitive proof is still lacking.
13:50
"Exploring Numbers: Loops, Sequences, and Conjectures"
- The narrator found a counterexample, leading to the realization of the correct statement and subsequent proof a month later.
- A plot of seed numbers up to 10,000 shows the largest number reached for each seed, with some numbers climbing as high as 27 million, indicating the potential for numbers to shoot off to infinity.
- Negative numbers exhibit three independent loops under the 3x+1 rules, starting at low values like -17 and -5, posing a question about the existence of disconnected loops on the negative side of the number line.
- Terry Tao's proof supports the conjecture that almost all numbers have a number in their sequence that is arbitrarily small, but this doesn't prove that all numbers follow this criteria.
- The percentage of perfect squares decreases as numbers increase, with almost all numbers not being perfect squares as the limit approaches infinity.
- The Collatz conjecture has been tested up to two to the 68, with no counterexample found yet, highlighting the vastness of the search space and the potential undecidability of the problem.
