# The Simplest Math Problem No One Can Solve - Collatz Conjecture

TLDRThe Collatz Conjecture, also known as 3x+1, is a simple yet unsolved problem in mathematics where a number undergoes specific operations based on its parity, eventually leading to a loop of 4, 2, 1. Despite its simplicity, the conjecture remains unproven, with no known counterexamples, and has been tested up to extraordinarily large numbers. The video explores various analyses, including geometric Brownian motion, Benford's law, and the halting problem, suggesting the possibility that the conjecture may be undecidable or that a counterexample exists beyond current computational limits.

### Takeaways

- 🧩 The Collatz Conjecture, also known as 3N+1, is an unsolved problem in mathematics where a sequence of operations on a number, involving multiplication by three and addition or division by two, supposedly always leads to the number one.
- 🔢 It is considered dangerous for young mathematicians to focus on this problem, as it has remained unsolved despite its simplicity, with renowned mathematician Paul Erdos suggesting mathematics is not yet ready for it.
- ⚔️ The conjecture's process involves applying two rules: if a number is odd, multiply by three and add one; if it's even, divide by two, and repeat until the sequence reaches one.
- 🌧️ The numbers generated by the 3x+1 process are called hailstone numbers, due to their unpredictable and fluctuating paths that eventually descend to one, similar to hailstones in a thundercloud.
- 📊 The paths of hailstone numbers can be visualized through graphs, revealing patterns that resemble geometric Brownian motion, which is also seen in the stock market's fluctuations.
- 📊 By examining the leading digits of hailstone numbers, a pattern emerges that aligns with Benford's Law, a distribution often found in naturally occurring data and used to detect anomalies or fraud.
- 📉 Despite the seemingly random nature of the sequences, mathematical analysis suggests that sequences are statistically more likely to shrink than grow due to the operations involved.
- 🌐 A directed graph of the 3x+1 sequences shows a complex network where every number is theorized to eventually connect back to the loop of four, two, one.
- 🔍 The conjecture could be false if a sequence is found that diverges to infinity or forms a closed loop disconnected from the main graph, but no such sequence has been found despite extensive testing.
- 🔢 Mathematicians have tested the conjecture up to very large numbers (two to the 68), and while this provides strong evidence for its validity, it does not constitute a formal proof.
- 🤔 The difficulty in proving the Collatz Conjecture may suggest that it is either exceptionally complex, potentially false, or possibly undecidable, reflecting the inherent challenges and unpredictability in the field of mathematics.

### Q & A

### What is the Collatz Conjecture?

-The Collatz Conjecture, also known as 3N+1, is a mathematical proposition that suggests that no matter what positive integer you start with, if you apply two simple rules—multiply by three and add one for odd numbers, and divide by two for even numbers—you will eventually reach the number one.

### Who is Paul Erdos, and what did he say about the Collatz Conjecture?

-Paul Erdos was a renowned mathematician known for his work in number theory. He is quoted as saying, 'Mathematics is not yet ripe enough for such questions,' implying that the field of mathematics may not have the necessary tools or understanding to solve the Collatz Conjecture at the time.

### What are hailstone numbers in the context of the Collatz Conjecture?

-Hailstone numbers are the numbers generated by applying the rules of the Collatz Conjecture. They fluctuate, sometimes rising to high values and then falling, similar to the movement of hailstones in a thundercloud, before eventually descending to one.

### What is the significance of the number 27 in the script?

-The number 27 is highlighted in the script as an example of a number that, when subjected to the Collatz Conjecture, reaches an unusually high peak of 9,232 before finally descending to one, demonstrating the unpredictable nature of the conjecture's sequences.

### What is the concept of 'stopping time' in relation to the Collatz Conjecture?

-The 'stopping time' of a number in the context of the Collatz Conjecture is the total number of steps it takes for the number to reduce to one after applying the conjecture's rules repeatedly.

### What is the difference between the paths taken by numbers in the Collatz Conjecture?

-The paths taken by numbers in the Collatz Conjecture can vary widely, even for numbers right next to each other. Some sequences may reach high peaks before descending, while others may do so more directly, reflecting the randomness and complexity of the conjecture.

### What is Benford's law, and how is it related to the Collatz Conjecture?

-Benford's law is a statistical principle that describes the frequency distribution of the first digits in many real-life sets of numerical data. In the context of the Collatz Conjecture, it is observed that the leading digits of hailstone numbers follow a pattern similar to Benford's law, with '1' being the most common leading digit.

### What does it mean for a sequence to be 'Turing-complete' as mentioned in the script?

-A sequence or system being 'Turing-complete' means that it has the ability to simulate the logic of a Turing machine, which is a theoretical model of computation. It can perform any computation that can be described through an algorithm, but it is also subject to the halting problem, which questions whether a computation will ever stop.

### What is the halting problem, and how does it relate to the Collatz Conjecture?

-The halting problem is a fundamental question in the theory of computation that asks whether it is possible to determine if a given program will finish running or continue to run forever. It is related to the Collatz Conjecture in the sense that if the conjecture were subject to the halting problem, it might be undecidable, meaning we may never be able to definitively prove or disprove it.

### What is the significance of the number 341 in the script?

-The number 341 is used in the script as an example to illustrate how quickly a large number can be reduced to one through the application of the Collatz Conjecture's rules, emphasizing the conjecture's potential to consistently lead to a reduction in sequence values.

### How have mathematicians attempted to prove the Collatz Conjecture using scatterplots?

-Mathematicians have used scatterplots to visualize the relationship between seed numbers and the numbers in their sequences. The idea is to show that in every 3x+1 sequence, there is at least one number smaller than the original seed, which would prove the conjecture by demonstrating that all sequences inevitably decrease to one.

### Outlines

### 🔢 The Enigma of the Collatz Conjecture

The Collatz Conjecture, also known as 3N+1, is a simple yet unsolved problem in mathematics that challenges even the best mathematicians. The conjecture suggests that any positive integer, when repeatedly subjected to the operations of multiplication by three and addition of one if it's odd, or division by two if it's even, will eventually reach the cycle of four, two, one, and then one. Despite its simplicity, the conjecture remains unproven, with Paul Erdos suggesting that mathematics is not yet ready for such questions. The process is illustrated with examples, and the unpredictability of the 'hailstone numbers' is highlighted, showing how numbers can rise and fall dramatically before converging to one. The video also touches on the various names and origins of the problem and the difficulty mathematicians face in making progress on it.

### 📊 Analyzing Patterns in 3x+1 Sequences

This section delves into the analysis of the 3x+1 problem through different mathematical lenses. It discusses the concept of 'hailstone numbers' and their erratic paths, which can reach astonishing heights before descending to one. The video introduces the idea of 'stopping time' and the randomness observed in the sequences, drawing parallels with geometric Brownian motion seen in stock markets. It also explores the leading digit phenomenon, known as Benford's law, which shows a predictable distribution of first digits in a wide range of numerical data, including those generated by the 3x+1 sequences. However, Benford's law does not provide a solution to the conjecture. The section also explains the statistical tendency of sequences to shrink rather than grow due to the mathematical operations involved and introduces the concept of a directed graph to visualize the connections between numbers in the sequence.

### 🔍 The Search for Counterexamples and Infinite Loops

The video continues by exploring the potential falsification of the Collatz Conjecture, which could occur if a number or sequence is found that either grows to infinity or forms a closed loop, disconnected from the main sequence. It mentions the extensive testing of numbers up to two to the 68th power, with no counterexamples found, and discusses the implications of this for the conjecture's validity. The video also presents various mathematical approaches to prove the conjecture, including scatterplots and the use of functions to show that almost all numbers in a sequence will eventually become smaller than the seed number. It highlights the work of mathematicians like Riho Terras and Terry Tao in bringing the problem closer to a resolution without fully solving it.

### 🤔 The Complexity of Numbers and the Limits of Proof

This section contemplates the nature of numbers and the peculiarities they exhibit, especially in the context of the Collatz Conjecture. It raises the question of whether the conjecture is difficult to prove because it might not be true, suggesting that more effort should be spent searching for counterexamples. The video contrasts the known behavior of perfect squares with the conjecture's statistical tendencies and the difficulty of proving absolutes in mathematics. It also discusses the Turing completeness of a related mathematical machine, FRACTRAN, and its connection to the halting problem, hinting at the possibility that the Collatz Conjecture may be undecidable.

### 🎨 The Aesthetic and Philosophical Implications of 3x+1

The final paragraph reflects on the beauty and complexity that arises from simple mathematical operations, as seen in the coral-like structures created by the 3x+1 sequences. It ponders the philosophical question of whether all numbers connect to this structure or if there exists a unique, unconnected path to infinity. The video concludes by acknowledging the difficulty of the problem and the limitations of current mathematical knowledge, as expressed by Paul Erdos. It also promotes the value of exploring and understanding mathematical problems through interactive learning platforms like Brilliant, which is sponsoring the video.

### Mindmap

### Keywords

### 💡Collatz Conjecture

### 💡Hailstone numbers

### 💡Geometric Brownian motion

### 💡Benford's law

### 💡Terry Tao

### 💡3x+1

### 💡Directed graph

### 💡Turing machine

### 💡Halting problem

### 💡FRACTRAN

### 💡Brilliant

### Highlights

The Collatz Conjecture, also known as 3N+1, is an unsolved problem in mathematics where a sequence of numbers is transformed by simple rules.

Paul Erdos suggested that mathematics may not yet be ready to solve the Collatz Conjecture.

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

The conjecture posits that any positive integer will eventually enter the 4-2-1 loop and reach one.

The conjecture is named after German mathematician Luther Collatz, who may have formulated it in the 1930s.

Numbers generated by the 3x+1 process are called hailstone numbers, due to their unpredictable rise and fall.

The stopping time of a number is the total steps it takes to reach one in the Collatz sequence.

The paths of hailstone numbers are highly irregular, even for numbers in close proximity.

Some mathematicians believe the 3x+1 problem was a Soviet invention to hinder U.S. scientific progress.

Jeffrey Lagarias is considered an authority on the 3x+1 problem, advising against dedicating a career to it.

Researchers have found that the sequences of hailstone numbers exhibit a pattern of randomness similar to geometric Brownian motion.

The leading digits of hailstone numbers follow Benford's law, a distribution common in various natural and financial phenomena.

Despite extensive testing of numbers up to two to the 68, no counterexample to the conjecture has been found.

Terry Tao demonstrated that almost all numbers in a 3x+1 sequence will eventually be smaller than the original seed.

The difficulty in proving the Collatz Conjecture may lie in its potential undecidability, similar to the halting problem in computer science.

The conjecture's truth is still uncertain, as all attempts to prove it have been unsuccessful, and counterexamples are yet to be found.

The inclusion of negative numbers in the 3x+1 process reveals additional loops, suggesting a complexity not present in the positive integers.

The Collatz Conjecture challenges traditional views of numbers as regular and patterned, highlighting their peculiar and unpredictable nature.

The problem's simplicity and the complexity of its implications make the Collatz Conjecture a compelling subject for mathematical exploration.