Самая простая нерешённая задача — гипотеза Коллатца [Veritasium]
Vert Dider・2 minutes read
The Collatz Conjecture involves a number choosing algorithm that eventually leads to the sequence 4-2-1, with numbers decreasing unpredictably and following Benford's law. Mathematical advancements suggest most numbers will be less than any function of X as X approaches infinity, but the hypothesis remains unproven, with the search for counterexamples crucial.
Insights
- The Collatz conjecture involves transforming numbers by multiplying odd ones by 3 and adding 1, and dividing even numbers by 2, eventually leading all positive integers to the cycle 4-2-1, showcasing the unpredictability and convergence of the algorithm.
- Mathematical advancements and attempts to prove the hypothesis have revealed the intricate nature of the Collatz conjecture, with the search for counterexamples being crucial, highlighting the complexity and unpredictability of numbers in challenging our understanding of mathematics.
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Recent questions
What is the Collatz conjecture?
The Collatz conjecture involves choosing numbers and applying specific rules to create a sequence.
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Summary
00:00
"The Collatz Conjecture: Unproven Mathematical Mystery"
- The hypothesis involves choosing any number, with two rules: if odd, multiply by 3 and add 1, if even, divide by 2.
- The hypothesis states that any positive integer following the algorithm will fall into the cycle 4-2-1, named after Lothar Collatz.
- Numbers obtained during transformations are called hail numbers, eventually falling to one.
- The path of numbers in the algorithm is unpredictable, resembling geometric Brownian movement.
- The distribution of first digits in sequences of hailstones follows Benford's law, common in various phenomena.
- Odd numbers in the algorithm don't increase as much as expected due to the division by 2 after multiplication.
- Visualization of the algorithm through oriented graphs shows a loop of numbers converging to 4-2-1.
- Attempts to prove the hypothesis include scatterplots and showing that almost all sequences have values lower than the original number.
- Recent mathematical advancements suggest that almost all numbers will be less than any function of X as X tends to infinity.
- The hypothesis remains unproven, with the search for counterexamples being crucial to either confirm or refute it.
16:58
Vast Numbers, Turing Machines, Mathematical Mysteries
- 10 to the power of 461 surpasses all numbers tested by the well hypothesis by 10,340 powers, illustrating a vast difference in magnitude. This problem can be likened to a program on a Turing machine, where the original number serves as input data, represented by a tape of 68 squares. While finding numbers meeting specific criteria in the algorithm is feasible, the search for counterexamples is impractical due to the infinite set of numbers, making manual calculation of 2000 powers unattainable.
- In 1987, John Conway introduced a generalization of the 3x + 1 problem through the fragtran mathematical machine, proving its Turing completeness akin to a modern computer. However, this also highlights the stalling problem where endless calculations occur without yielding results, leaving the well hypothesis unconfirmed. Despite the belief in mathematical certainty, the complexity and unpredictability of numbers challenge our understanding, emphasizing the intricate nature of mathematics and the mysteries it still holds.
