The Oldest Unsolved Problem in Math
TLDRThis video explores the oldest unsolved problem in mathematics: the existence of odd perfect numbers. It traces the historical quest to understand perfect numbers, discusses Euclid's and Euler's contributions, and highlights modern computational efforts like GIMPS. The video also ponders the potential non-existence of odd perfect numbers and the philosophical value of pursuing unsolved mathematical problems.
Takeaways
- ๐ The oldest unsolved problem in math is whether odd perfect numbers exist, a question that has puzzled mathematicians for over 2000 years.
- ๐ A perfect number is one where the sum of its proper divisors equals the number itself, such as 6 (1+2+3=6) and 28 (1+2+4+7+14=28).
- ๐งฉ All known perfect numbers are even and follow a specific pattern related to triangular numbers and sums of consecutive odd cubes.
- ๐ Euclid discovered a formula for generating even perfect numbers, which involves multiplying a prime number by the sum of consecutive powers of 2.
- ๐ค Nicomachus's conjectures, which included that all perfect numbers are even and end in 6 or 8 alternately, were largely disproven over time.
- ๐ The search for perfect numbers has been aided by computers, with the Great Internet Mersenne Prime Search (GIMPS) project being particularly successful in finding new Mersenne primes and corresponding perfect numbers.
- ๐ The discovery of a new Mersenne prime can earn recognition and even a cash prize, highlighting the ongoing interest and competition in this field.
- ๐ The largest known Mersenne prime, and thus perfect number, is incredibly large, with the 51st Mersenne Prime having over 24 million digits.
- ๐ Despite extensive research, no odd perfect numbers have been found, and recent progress suggests that if they exist, they must be larger than 10^2200.
- ๐ The pursuit of perfect numbers, while seemingly abstract, has historically led to significant advancements in number theory and could have unforeseen applications in the future.
Q & A
What is the oldest unsolved problem in math mentioned in the video?
-The oldest unsolved problem in math mentioned in the video 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 the number itself. For example, the number 6 is a perfect number because its proper divisors (1, 2, and 3) add up to 6.
What is the significance of the number 6 in the context of perfect numbers?
-The number 6 is significant because it is the first perfect number. Its proper divisors (1, 2, and 3) add up to 6, which is the number itself.
What is the pattern that Euclid discovered for generating even perfect numbers?
-Euclid discovered that even perfect numbers can be generated using the formula 2^(p-1) * (2^p - 1), where 2^p - 1 is a prime number (Mersenne prime).
What is the connection between Mersenne primes and perfect numbers?
-Mersenne primes are prime numbers of the form 2^p - 1. When multiplied by 2^(p-1), they result in even perfect numbers, according to Euclid's formula.
What was the contribution of Leonhard Euler to the understanding of perfect numbers?
-Leonhard Euler made significant contributions by discovering the eighth perfect number, inventing the sigma function, proving that every even perfect number has Euclid's form (the Euclid-Euler theorem), and refining the form for odd perfect numbers.
What is the sigma function, and how is it used in the study of perfect numbers?
-The sigma function is a mathematical function that sums up all the divisors of a number, including the number itself. It is used to study perfect numbers because the sigma function of a perfect number always gives twice the number itself.
What is the Great Internet Mersenne Prime Search (GIMPS), and how does it contribute to the discovery of Mersenne primes?
-The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project that uses the distributed computing power of volunteers' computers to search for Mersenne primes. It has been successful in discovering new Mersenne primes and their corresponding perfect numbers.
What is the current status of the search for odd perfect numbers?
-As of the video's information, no odd perfect numbers have been found. Researchers have used algorithms to show that if an odd perfect number exists, it must be larger than 10^2200, making it highly unlikely to be found with current computational methods.
Why is the study of perfect numbers important, even if they have no known practical applications?
-The study of perfect numbers is important because it drives mathematical curiosity and can lead to the development of new mathematical theories and techniques. Even though they currently have no known practical applications, the foundational work in number theory has historically led to significant advancements in fields like cryptography.
Outlines
๐ข The Quest for Odd Perfect Numbers
This paragraph introduces the ancient and elusive problem of finding odd perfect numbers. A perfect number is one where the sum of its proper divisors equals the number itself, exemplified by the numbers 6, 28, 496, and 8,128. The speaker highlights the simplicity and beauty of the problem, which has intrigued mathematicians for over two millennia. Despite extensive computational efforts checking numbers up to 10^2,200, no odd perfect numbers have been discovered. The historical significance and the allure of the problem are emphasized, setting the stage for a deep dive into its mathematical intricacies.
๐ Euclid's Formula and the Nature of Perfect Numbers
This paragraph delves into the discovery and properties of perfect numbers, focusing on Euclid's formula for generating even perfect numbers. It explains how these numbers can be represented as the sum of consecutive powers of two, and how they are related to triangular numbers and sums of consecutive odd cubes. The historical context is provided, mentioning Nicomachus's conjectures and the later disproof of some by Egyptian mathematician Ibn Fallus. The paragraph also discusses the work of Marin Mersenne and his list of potential primes, which led to the discovery of additional perfect numbers.
๐ค Euler's Contributions to the Theory of Perfect Numbers
This paragraph highlights the significant contributions of Leonhard Euler to the understanding of perfect numbers. Euler discovered the eighth perfect number and introduced the sigma function, a tool for summing the divisors of a number. He used this function to prove that every even perfect number fits Euclid's form, thereby solving a 1600-year-old problem. Euler also explored the possibility of odd perfect numbers, suggesting they must have a specific form if they exist, but was unable to prove their existence or non-existence.
๐ The Great Internet Mersenne Prime Search (GIMPS)
This paragraph discusses the modern era of perfect number research, particularly the advent of the Great Internet Mersenne Prime Search (GIMPS). It explains how this collaborative project uses distributed computing to search for Mersenne primes, which correspond to perfect numbers. The paragraph details the discovery of the 50th Mersenne Prime and the cultural impact of such findings, including the publication of a book containing the prime number and the financial incentives for discovering new primes.
๐ The Search for Odd Perfect Numbers Continues
This paragraph explores the ongoing search for odd perfect numbers and the challenges involved. It mentions the heuristic argument made by Carl Pomerance, which suggests that the likelihood of finding odd perfect numbers is extremely low. The paragraph also discusses the concept of 'spoofs' โ numbers that appear to be perfect but are not โ and how researchers are using these to further understand the properties of potential odd perfect numbers. The speaker expresses a personal belief that odd perfect numbers do not exist, based on the current evidence and mathematical reasoning.
๐ The Value of Pursuing Mathematical Curiosities
This final paragraph reflects on the importance of pursuing mathematical problems like the search for perfect numbers, even if they have no immediate practical applications. The speaker argues that the pursuit of such problems can lead to unexpected breakthroughs and advancements in mathematics and technology. The paragraph concludes with a plug for the learning platform Brilliant, encouraging viewers to explore their curiosity and develop problem-solving skills through a wide range of subjects.
Mindmap
Keywords
๐กOdd Perfect Numbers
๐กProper Divisors
๐กTriangular Numbers
๐กEuclid's Formula
๐กMersenne Primes
๐กEuler's Sigma Function
๐กEuclid-Euler Theorem
๐กHeuristic Argument
๐กGreat Internet Mersenne Prime Search (GIMPS)
๐กCryptography
๐กSpoofs
Highlights
The oldest unsolved problem in mathematics is whether odd perfect numbers exist, a question that has puzzled mathematicians for over 2000 years.
Perfect numbers are rare and unique; they are equal to the sum of their proper divisors, exemplified by the numbers 6 and 28.
All known perfect numbers are even and can be derived using Euclid's formula, which involves prime numbers and powers of two.
Euclid's pattern for generating even perfect numbers has been a cornerstone in the search for odd perfect numbers.
Nicomachus's conjectures, which included the belief that all perfect numbers are even, were unproven for over a thousand years.
The discovery of the fifth perfect number disproved Nicomachus's first conjecture about the number of digits in perfect numbers.
Mersenne primes are crucial in identifying even perfect numbers, as they are the building blocks of Euclid's formula.
Leonhard Euler's contributions to number theory, including the sigma function, were instrumental in proving the Euclid-Euler theorem.
Euler's work suggested that odd perfect numbers, if they exist, must have a specific form involving prime numbers.
The search for odd perfect numbers has led to the establishment of lower bounds that are exponentially larger than previously thought.
The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project that has successfully discovered several new Mersenne primes.
The discovery of the 50th Mersenne Prime in 2017, a number with over 23 million digits, highlights the sheer scale of these numbers.
The Lenstra and Pomerance Wagstaff conjecture suggests the existence of infinitely many Mersenne primes, and thus even perfect numbers.
Despite the lack of practical applications, the study of perfect numbers has driven the development of new mathematical theories and methods.
The heuristic argument suggests that the likelihood of odd perfect numbers existing is extremely low, but it is not a definitive proof.
The pursuit of knowledge in mathematics, even in areas without immediate applications, can lead to significant discoveries and advancements.
Brilliant, the learning platform, encourages the development of problem-solving skills and curiosity in various fields, including mathematics.