The Simplest Math Problem No One Can Solve - Collatz Conjecture

Veritasium

30 Jul 202122:09

TLDRThe Collatz Conjecture, also known as 3x+1, is a simple yet unsolved problem in mathematics. It involves applying two rules to any positive integer: multiply by three and add one if the number is odd, or divide by two if it's even. The conjecture suggests all numbers will eventually reach a loop of four, two, one. Despite extensive testing and various analyses, including Benford's law and geometric Brownian motion, a proof remains elusive. Mathematicians like Terry Tao have shown that almost all numbers follow the conjecture, but proving it for all numbers remains a challenge, highlighting the complexity and peculiarity of numbers.

Takeaways

⚠️ The Collatz conjecture is a simple yet unsolved problem in mathematics that has baffled even the greatest mathematicians.
🔢 The conjecture involves picking any number and applying two rules: multiply by three and add one if the number is odd, and divide by two if the number is even.
🔄 The conjecture states that every positive integer will eventually end up in the cycle 4, 2, 1.
📈 The sequences generated by this problem, known as hailstone numbers, rise and fall like hailstones before eventually settling into the 4, 2, 1 loop.
🧠 Despite extensive efforts and brute-force testing of numbers up to 2^68, no counterexamples to the conjecture have been found.
📉 The pattern of the Collatz sequences appears random, resembling the behavior of the stock market, but it tends to decrease over time.
🔢 Mathematicians have discovered that almost all Collatz sequences reach a point below their initial value, but this has not been proven for all numbers.
📊 The leading digits of numbers in Collatz sequences follow Benford's law, a distribution also observed in many natural datasets.
🤔 Mathematicians have proposed various hypotheses and analyses, but none have definitively proven or disproven the Collatz conjecture.
💡 The Collatz conjecture may be undecidable, similar to other problems in theoretical computer science, leaving its truth forever unknown.

Q & A

What is the Collatz conjecture?
-The Collatz conjecture is a mathematical hypothesis that states that starting with any positive integer and applying the following rules—if the number is odd, multiply by three and add one; if the number is even, divide by two—will eventually lead to the number one.
Who is credited with the Collatz conjecture?
-The conjecture is named after the German mathematician Lothar Collatz, who may have come up with it in the 1930s.
Why is the Collatz conjecture considered difficult to solve?
-Despite its simple formulation, no one has been able to prove or disprove the conjecture. The paths that numbers take under the rules of the conjecture appear random and unpredictable, making it challenging to establish a general proof.
What did Paul Erdos say about the Collatz conjecture?
-Paul Erdos, a famous mathematician, said, 'Mathematics is not yet ripe enough for such questions,' indicating the difficulty and complexity of the problem.
What are 'hailstone numbers'?
-'Hailstone numbers' refer to the sequence of numbers generated by applying the 3x+1 rules to a starting integer. These numbers are called hailstone numbers because they can rise and fall in value like hailstones in a thundercloud before eventually falling to one.
What notable mathematician warned against working on the Collatz conjecture?
-Jeffrey Lagarias is a noted mathematician who warned a younger mathematician, Alex Kontorovich, against working on the Collatz conjecture, suggesting it could be a career risk.
What is the significance of the number 27 in the Collatz conjecture?
-Starting with the number 27, the sequence climbs as high as 9,232 before eventually falling back down to one. This showcases the erratic behavior of numbers under the Collatz rules.
What does the scatterplot analysis of the Collatz conjecture aim to prove?
-The scatterplot analysis aims to show that in every 3x+1 sequence, there is a number that is smaller than the original seed, which would help in proving the Collatz conjecture by showing that every sequence eventually falls to one.
What was Terry Tao's contribution to the Collatz conjecture?
-In 2019, Terry Tao showed that almost all numbers in the Collatz sequences will end up smaller than any arbitrary function f(x), so long as that function goes to infinity as x goes to infinity. This brought mathematicians closer to proving the conjecture but did not provide a complete proof.
What are some alternative names for the Collatz conjecture?
-The Collatz conjecture is also known as the Ulam conjecture, Kakutani's problem, Thwaites conjecture, Hasse's algorithm, the Syracuse problem, and simply 3N+1.
Why might the Collatz conjecture be considered undecidable?
-John Conway created a generalization of the Collatz conjecture, called FRACTRAN, and showed it is Turing-complete. This suggests it could be subject to the halting problem, implying it may be undecidable whether the conjecture is true or false.

Outlines

00:00

🔢 The Enigma of the Collatz Conjecture

The Collatz Conjecture, also known as 3N+1, is introduced as a seemingly simple yet unsolved problem in mathematics. It involves a sequence where numbers are transformed based on whether they are odd or even. The conjecture posits that any positive integer will eventually enter the '4, 2, 1' cycle and reach one. Despite its simplicity, this problem has eluded proof, with mathematicians like Paul Erdős suggesting mathematics is not yet ready for such questions. The video explains the process of the conjecture with examples and discusses its various names and the randomness observed in the sequences, comparing it to geometric Brownian motion.

05:00

📊 Patterns in the 3x+1 Sequences

This section delves into the analysis of the 3x+1 problem, exploring the leading digits of numbers in the sequences and the emergence of Benford's Law, which describes the distribution of these digits across various datasets. The video discusses the peculiarity that while the conjecture suggests sequences should shrink, odd numbers are effectively increased by a factor of 3/2, leading to an overall trend of reduction. The use of directed graphs to visualize the paths of numbers in the sequence is introduced, along with the concept of geometric mean to explain the statistical likelihood of sequences shrinking. The possibility of the conjecture being false due to a divergent sequence or a closed loop is also considered.

10:02

🔍 The Search for Counterexamples and Proofs

The video outlines the extensive efforts to prove or disprove the Collatz Conjecture, including testing every number up to 2^68, which equates to nearly 300 quintillion numbers. It discusses the statistical approaches by mathematicians like Riho Terras and Terry Tao, who have shown that almost all numbers in a 3x+1 sequence will eventually become smaller than their original seed, bringing them closer to proving the conjecture but not quite achieving it. The video also raises the question of whether the conjecture might be false and the importance of looking for counterexamples, as well as the challenges in exhaustively searching the infinite space of possibilities.

15:03

🤔 The Philosophical and Practical Implications

This part of the script ponders the philosophical implications of the Collatz Conjecture, questioning the nature of mathematical truth and the possibility that the conjecture may be undecidable, similar to the halting problem in computer science. It contrasts the simplicity of the conjecture with the complexity of the patterns it generates, such as the coral-like structures when visualized. The video also reflects on the peculiarity of numbers and the difficulty in proving theorems that may be false, suggesting that the struggle to prove the conjecture might indicate its falsity.

20:04

🌐 The Beauty of Mathematics and the 3x+1 Problem

The final paragraph transitions to a discussion on the beauty and complexity of mathematics, as exemplified by the 3x+1 problem. It emphasizes the importance of understanding and exploring mathematical problems through interactive learning, promoting the use of Brilliant.org for engaging with mathematical concepts. The video ends with a call to action for viewers to join the learning community and explore mathematical fundamentals and algorithm courses, sponsored by Brilliant, and provides a link for further exploration.

Mindmap

Keywords

💡Collatz Conjecture

The Collatz Conjecture, also known as the 3N+1 problem, is a mathematical hypothesis that concerns a sequence defined for any positive integer. The conjecture posits that no matter what number you start with, if you repeatedly apply the rules of multiplying by three and adding one for odd numbers and dividing by two for even numbers, you will eventually reach the number one. In the video, the Collatz Conjecture is presented as a simple yet unsolved problem that highlights the complexity and unpredictability within seemingly straightforward mathematical operations.

💡Hailstone numbers

Hailstone numbers are the terms in the sequence generated by the Collatz Conjecture. The name 'hailstone' is metaphorical, as the numbers bounce up and down like hailstones in a thundercloud before inevitably falling to the ground, which in this case means reaching the number one. The video script uses the term to describe the unpredictable paths these numbers take before converging to one, emphasizing the erratic behavior of the sequence.

💡Geometric Brownian motion

Geometric Brownian motion is a mathematical model used to represent the random fluctuations of a variable, such as stock prices, over time. In the context of the video, the Collatz sequence's logarithmic graph is compared to geometric Brownian motion, illustrating the seemingly random pattern of the sequence's growth and decline, akin to the unpredictable movement of financial markets.

💡Benford's law

Benford's law is a principle in probability that predicts the frequency distribution of the first digits in many real-life sets of numerical data. The video explains that when analyzing the leading digits of hailstone numbers, a pattern emerges that aligns with Benford's law, where lower digits are more common as leading digits. This observation is used to further explore the statistical properties of the Collatz sequence.

💡Terry Tao

Terry Tao is a renowned mathematician mentioned in the video for his significant contributions to the study of the Collatz Conjecture. His work has shown that almost all numbers in the sequence will eventually become smaller than the original seed number, bringing the mathematical community closer to proving the conjecture, although a definitive proof remains elusive.

💡3x+1

The term '3x+1' is a shorthand reference to the Collatz Conjecture, derived from the rule applied to odd numbers in the sequence. The video script uses this term to denote the specific operation that, when applied to a number, is a key part of the sequence's progression towards one. It is also used to discuss the conjecture's potential truth and the difficulty in proving it.

💡Directed graph

A directed graph is a graphical representation where edges have a direction, indicating a relationship between two vertices. In the video, the paths of numbers in the Collatz sequence are visualized as a directed graph, symbolizing the flow of numbers from one step to the next, potentially leading to the conjecture's proof or revealing a counterexample.

💡Turing machine

A Turing machine is a theoretical computational model that defines an algorithm's logic. The video script likens the Collatz Conjecture to a simple program run on a Turing machine, where the seed number is the input. This analogy is used to discuss the potential undecidability of the conjecture, referencing the halting problem associated with Turing machines.

💡Counterexample

A counterexample is a particular instance or case that disproves a general statement or theory. The video discusses the possibility that a counterexample could exist for the Collatz Conjecture, which would disprove the conjecture. The search for such a counterexample is part of the ongoing effort to resolve the conjecture.

💡Undecidable

In mathematics and logic, an undecidable problem is one for which no solution or proof can be found within a given system or framework. The video raises the possibility that the Collatz Conjecture might be undecidable, suggesting that despite extensive testing and analysis, it may never be proven or disproven.

💡FRACTRAN

FRACTRAN is a mathematical model created by John Conway, which is a generalization of the Collatz Conjecture. The video mentions FRACTRAN to illustrate the potential complexity and computational challenges associated with the conjecture, as FRACTRAN's Turing-completeness implies it can simulate any computation, including those that may never halt.

Highlights

The Collatz Conjecture, also known as 3x+1, is an unsolved problem in mathematics where sequences of numbers, starting from any positive integer, are transformed following simple rules and are conjectured to always reach the cycle of 4, 2, 1.

Paul Erdos suggested that mathematics is not yet ready for such questions, indicating the complexity and the depth of the problem.

The conjecture's process involves multiplying an odd number by three and adding one, or dividing an even number by two, and repeating these steps indefinitely.

The sequence's behavior is unpredictable, with numbers like 27 reaching heights comparable to Mount Everest before descending to one.

Hailstone numbers, derived from the 3x+1 process, exhibit a randomness akin to geometric Brownian motion, similar to stock market fluctuations.

Despite the apparent randomness, the leading digits of hailstone numbers follow Benford's Law, a distribution observed in various natural and financial phenomena.

The conjecture's difficulty lies in the fact that while sequences statistically tend to shrink, there's no definitive proof that they cannot grow indefinitely for some starting numbers.

Terry Tao, a renowned mathematician, has shown that almost all numbers in a 3x+1 sequence will eventually become smaller than the original seed, bringing the problem closer to a solution without proving it.

The conjecture's validity has been tested for numbers up to 2^68, a number so large it dwarfs the combined total of all other tested numbers.

The possibility of a counterexample existing, one that defies the conjecture and grows to infinity, remains open and could potentially disprove the conjecture.

The inclusion of negative numbers in the 3x+1 process reveals additional loops, suggesting a disparity between the behavior of positive and negative integers.

The conjecture's proof eludes mathematicians, possibly due to its inherent complexity or the possibility that it may not be universally true.

The halting problem, related to the Turing machine's inability to determine whether a computation will finish, casts doubt on the possibility of ever proving the conjecture.

The video emphasizes the peculiar nature of numbers, challenging the perception of their regularity and predictability, as seen in the organic patterns of the coral representation of 3x+1 sequences.

The Collatz Conjecture exemplifies the limitations of human understanding in mathematics, where even simple problems can resist solution despite extensive computational testing.

The video concludes by highlighting the beauty and mystery of mathematics, where problems like the Collatz Conjecture underscore the vastness of unsolved questions.

Brilliant, the sponsor of the video, is promoted as a resource for interactive learning and problem-solving, aiming to inspire daily learning and understanding in mathematics and related fields.

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