The Oldest Unsolved Problem in Math

Veritasium

7 Mar 202431:32

EducationalLearning

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TLDRThe video delves into the enigmatic world of perfect numbers, exploring their history, properties, and the enduring mathematical quest to uncover both even and odd perfect numbers. It highlights the contributions of mathematicians like Euclid, Descartes, and Euler, and the ongoing search for the elusive odd perfect number, which remains an unsolved problem. The discussion also touches on the significance of pure mathematical curiosity and its potential to lead to practical applications, as seen in the development of cryptography from number theory.

Takeaways

📚 The oldest unsolved problem in math is whether odd perfect numbers exist, a mystery that dates back 2000 years.
🧠 Mathematicians have used computers to check numbers up to 10^2200, but have not found any odd perfect numbers yet.
🤔 The allure of this problem lies in its simplicity, beauty, and the historical challenge it poses to the greatest minds.
📈 Perfect numbers are defined as numbers where the sum of their proper divisors (excluding the number itself) equals the number.
🔢 Known perfect numbers are 6, 28, 496, and 8128, all of which are even and follow a specific pattern related to Mersenne primes.
🔄 The pattern of even perfect numbers was discovered by Euclid and involves consecutive powers of 2 and Mersenne primes.
🧩 Each even perfect number can be expressed as 2^(p-1) * (2^p - 1), where p is prime, known as Euclid's formula for even perfect numbers.
🔢 The conjecture by Nicomachus that all even perfect numbers follow Euclid's form was proven true by Euler, known as the Euclid-Euler theorem.
💡 Euler also made significant progress on the problem of odd perfect numbers, setting up conditions that any odd perfect number must satisfy.
🌐 The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project that has discovered numerous Mersenne primes and corresponding perfect numbers.
🚀 The search for perfect numbers has been aided by advanced computational methods, but the existence of odd perfect numbers remains unproven.

Q & A

What is the oldest unsolved problem in math mentioned in the video?
-The oldest unsolved problem mentioned is whether any odd perfect numbers exist.
What is a perfect number?
-A perfect number is a positive integer that is equal to the sum of its proper divisors, excluding itself.
How have mathematicians attempted to find a pattern in perfect numbers?
-Mathematicians have noticed that perfect numbers tend to be one digit longer than the previous one, and that they alternate between ending in 6 and 8. They are also all even and can be expressed as the sum of consecutive numbers, creating a triangular pattern.
What is Euclid's formula for generating even perfect numbers?
-Euclid's formula states that an even perfect number can be expressed as 2^(p-1) * (2^p - 1), where (2^p - 1) is prime.
What is a Mersenne Prime and how is it related to perfect numbers?
-A Mersenne Prime is a prime number that can be written in the form 2^p - 1, where p is also a prime number. Mersenne Primes are related to perfect numbers because if 2^p - 1 is prime, then 2^(p-1) * (2^p - 1) is a perfect number.
What is the Great Internet Mersenne Prime Search (GIMPS)?
-The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project that uses distributed computing to search for Mersenne primes, with volunteers donating their computer's processing power to the cause.
What is a 'spoof' in the context of odd perfect numbers?
-A 'spoof' is a number that closely resembles an odd perfect number but is not actually perfect. It shares many properties with odd perfect numbers but lacks the additional properties required for a number to be truly perfect.
What is the significance of the number 22021 in the context of the video?
-22021 was considered a potential factor in an equation that could have represented an odd perfect number. However, it turned out not to be prime, and thus the number in question was not perfect.
What is the current lower bound for the existence of an odd perfect number?
-The current lower bound for the existence of an odd perfect number is larger than 10 to the power of 2,200.
What is the Lenstra and Pomerance Wagstaff conjecture?
-The Lenstra and Pomerance Wagstaff conjecture predicts the density of Mersenne primes based on the size of p, suggesting that there are infinitely many Mersenne primes, and therefore infinitely many even perfect numbers.
How does the video relate the study of perfect numbers to the development of modern cryptography?
-The video suggests that although number theory, including the study of perfect numbers, had no known real-world applications for over 2000 years, the foundational knowledge it developed eventually became the basis for modern cryptography, which is crucial for securing digital communications and protecting sensitive information.
What lesson does the video convey about pursuing mathematical problems without immediate practical applications?
-The video emphasizes the value of curiosity-driven research, suggesting that even seemingly 'useless' mathematical problems can lead to significant advancements and practical applications in the future, as was the case with number theory and cryptography.

Outlines

00:00

🧩 The Enigma of Odd Perfect Numbers

This paragraph delves into the ancient and captivating problem of odd perfect numbers. It introduces the concept of perfect numbers, which are numbers that equal the sum of their proper divisors, excluding themselves. The discussion highlights the historical quest to find odd perfect numbers, which has stumped mathematicians for over 2000 years. The paragraph also touches on the significance of this problem in the realm of mathematics and the allure it holds for mathematicians due to its simplicity and beauty.

05:01

📚 Euclid's Legacy and the Quest for Patterns

This section explores the historical attempts to find patterns within perfect numbers and the mathematical breakthroughs in understanding them. It discusses Euclid's formula for generating even perfect numbers and the conjectures of Nicomachus, which were widely accepted for centuries. The paragraph also covers the debunking of these conjectures and the evolution of the understanding of perfect numbers, including the work of mathematicians like Ibn Fallus, Marin Mersenne, Pierre de Fermat, and René Descartes.

10:01

🦅 Euler's Contributions and the Sigma Function

This paragraph focuses on the significant contributions of Leonhard Euler to the study of perfect numbers. It explains Euler's discovery of the eighth perfect number and his invention of the sigma function, a powerful mathematical tool for analyzing the divisors of a number. The section details how Euler used the sigma function to prove the Euclid-Euler theorem, which solidified the understanding of even perfect numbers, and his attempts to solve the riddle of odd perfect numbers.

15:04

🚀 The Modern Pursuit of Mersenne Primes

This section discusses the modern advancements in the search for Mersenne primes and the corresponding perfect numbers. It describes the work of Edouard Lucas and Frank Nelson Cole, the development of computer programs to check for prime numbers, and the establishment of the Great Internet Mersenne Prime Search (GIMPS). The paragraph also highlights the sheer size of the largest known primes and the collaborative effort of the mathematical community in the ongoing search for more perfect numbers.

20:05

📈 The Heuristic Approach and the Future

This paragraph examines the heuristic approach to the problem of odd perfect numbers, which, while not providing a proof, offers a statistical argument against their existence. It discusses the predictions made by Carl Pomerance and the ongoing efforts to refine these heuristics. The section also explores the philosophical aspects of pursuing mathematical problems without immediate practical applications, emphasizing the historical importance of curiosity-driven research in number theory.

25:05

🌟 The Power of Curiosity and Learning

In this final paragraph, the focus shifts to the broader implications of studying unsolved mathematical problems. It argues for the value of following curiosity and the potential for such pursuits to lead to significant advancements in the future. The paragraph concludes with a message of encouragement for young mathematicians to engage with challenging problems and a call for a balance between practical skill-building and intellectual exploration.

Mindmap

Keywords

💡Perfect Numbers

Perfect numbers are a unique set of numbers where the sum of their proper divisors (excluding the number itself) equals the number. For example, the number 6 is perfect because 1, 2, and 3 add up to 6. They have fascinated mathematicians for centuries due to their rarity and the elegance of their properties. The video discusses the history and mystery surrounding these numbers, particularly the question of whether odd perfect numbers exist.

💡Mersenne Primes

Mersenne primes are a special class of prime numbers that can be written in the form 2^p - 1, where p itself is a prime number. They are significant in the context of perfect numbers because they are used to generate even perfect numbers through a formula. The video discusses the work of Marin Mersenne and others in identifying these primes and their relation to the discovery of new perfect numbers.

💡Euclid's Formula

Euclid's formula is a method for generating even perfect numbers, discovered by the ancient Greek mathematician Euclid. It states that every even perfect number can be expressed as 2^(p-1) multiplied by (2^p - 1), where 2^p - 1 is a Mersenne prime. This formula has been instrumental in the search for new perfect numbers and in understanding their structure.

💡Odd Perfect Numbers

Odd perfect numbers are a hypothetical set of numbers that are not even and for which the sum of their proper divisors equals the number itself. Unlike even perfect numbers, no odd perfect number has been found, and their existence remains an open question in mathematics. The video discusses the search for these numbers and the challenges involved in proving or disproving their existence.

💡Nicomachus's Conjectures

Nicomachus's conjectures are a set of five statements about perfect numbers proposed by the ancient Greek mathematician Nicomachus. These conjectures include claims that all perfect numbers are even, end alternately in 6 and 8, and that there are infinitely many perfect numbers. The video discusses how these conjectures have guided the study of perfect numbers for centuries, some of which have been proven false while others remain unproven.

💡Euler's Contributions

Leonhard Euler, a prominent mathematician, made significant contributions to the study of perfect numbers. He discovered an eighth perfect number and invented the sigma function, which helps in understanding the properties of perfect numbers. Euler also proved that every even perfect number must have Euclid's form, confirming Nicomachus's fourth conjecture. His work brought the study of perfect numbers to a new level and laid the groundwork for further research.

💡Great Internet Mersenne Prime Search (GIMPS)

GIMPS is a collaborative project that uses distributed computing to search for Mersenne primes, which in turn can lead to the discovery of new perfect numbers. It allows volunteers to contribute their computer's processing power to the search. The project has been successful in finding several new Mersenne primes and has made significant contributions to the field of number theory.

💡Heuristic Arguments

Heuristic arguments are non-rigorous methods of reasoning that provide plausible reasons for believing a proposition to be true or false. In the context of the video, heuristic arguments are used to suggest that odd perfect numbers may not exist, based on the observed patterns and the rarity of primes of a certain type. However, these arguments do not constitute a formal proof.

💡Spoofs

Spoofs are numbers that almost meet the properties of being an odd perfect number but ultimately do not. They share many of the characteristics of odd perfect numbers but fail to meet all the necessary conditions. The study of spoofs helps mathematicians understand the properties that genuine odd perfect numbers would need to have and could potentially lead to a proof that odd perfect numbers cannot exist.

💡Binary Representation

The binary representation of a number is its expression in the base-2 numeral system, which uses only two symbols: 0 and 1. In the context of the video, the binary forms of known perfect numbers are discussed, revealing a pattern of consecutive powers of two, which is a fascinating property of these numbers and contributes to their mystique.

💡Cryptography

Cryptography is the practice of secure communication in the presence of adversaries. It involves the use of mathematical techniques to protect information from unauthorized access. The video mentions the application of number theory, which includes the study of perfect numbers, in the development of cryptographic methods that protect digital communications and sensitive data.

Highlights

The oldest unsolved problem in math dates back 2000 years, captivating the imaginations of many mathematicians due to its simplicity and beauty.

The problem revolves around the existence of odd perfect numbers, which, if found, would be as simple as identifying a single number.

Mathematicians have used computers to check numbers up to 10^2200, but have yet to find a solution.

A perfect number is defined by its divisors summing up to the number itself, such as 6 and 28.

Euclid discovered a pattern for even perfect numbers, which involves consecutive powers of two and primes.

Nicomachus made five conjectures about perfect numbers, which were considered facts for a thousand years until some were disproven.

Mersenne Primes are crucial for finding even perfect numbers, and the search for them has led to the discovery of new perfect numbers.

Euler made significant contributions to the understanding of perfect numbers, including the Euclid-Euler theorem.

The sigma function, invented by Euler, is a powerful tool for analyzing the divisors of a number and has been key in the study of perfect numbers.

Despite advancements, the existence of odd perfect numbers remains unproven, with researchers setting increasingly high lower bounds for their potential size.

The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project that has discovered several new Mersenne primes and perfect numbers.

The discovery of a new Mersenne prime can lead to the recognition of the discoverer and even a substantial monetary prize.

The search for Mersenne primes and perfect numbers has been aided by the rapid growth of computer technology.

The Lenstra and Pomerance Wagstaff conjecture suggests there are infinitely many Mersenne primes and even perfect numbers.

The concept of 'spoofs' has been introduced to describe numbers that almost meet the criteria for being odd perfect numbers.

Heuristic arguments suggest that odd perfect numbers may not exist, but this is not a formal proof.

The study of perfect numbers, despite its lack of direct applications, reflects the value of pure mathematical curiosity and the potential for future discoveries.

The pursuit of solving the problem of perfect numbers exemplifies the spirit of mathematical exploration and the potential for unexpected applications.

The video encourages young mathematicians to engage with challenging problems and emphasizes the importance of following one's curiosity in mathematical research.

Transcripts

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Perfect Numbers Number Theory Math History Euclid Mersenne Primes Odd Perfect Numbers Leonhard Euler Computational Math Brilliant Platform Educational