The Oldest Unsolved Problem in Math
Veritasium・24 minutes read
The oldest unsolved math problem, the existence of odd perfect numbers, continues to captivate mathematicians through history and into the modern era, showcasing the enduring pursuit of knowledge and potential future applications. Engaging with problems like perfect numbers can help develop problem-solving skills and critical thinking, with platforms like Brilliant offering a comprehensive learning experience that combines theoretical knowledge with practical applications in various fields.
Insights
- Perfect numbers are a subset of numbers where the sum of their divisors equals the number itself, with only a few known examples, leading mathematicians to explore their properties through various mathematical methods and computer calculations.
- The pursuit of solving the oldest math problem, odd perfect numbers, has not only led to advancements in number theory but also demonstrates the enduring appeal and collaborative nature of mathematical exploration, offering a valuable learning opportunity for students and enthusiasts to engage with complex problem-solving and critical thinking skills.
Get key ideas from YouTube videos. It’s free
Recent questions
What are perfect numbers?
Perfect numbers are numbers where the sum of their proper divisors equals the number itself. For example, 6 and 28 are perfect numbers because the sum of their divisors (1+2+3=6, 1+2+4+7+14=28) equals the number itself.
How are perfect numbers related to triangular numbers?
Perfect numbers are also triangular numbers, which means they can be represented as equilateral triangles. The relationship between perfect numbers and triangular numbers is a fascinating mathematical connection that showcases the beauty and complexity of number theory.
What is the significance of Mersenne Primes in relation to perfect numbers?
Mersenne Primes are closely linked to perfect numbers, as they correspond to the discovery of new perfect numbers. The search for Mersenne Primes has led to the identification of numerous perfect numbers, highlighting the intricate interplay between prime numbers and perfect numbers in the realm of mathematics.
How do Euler's sigma function and Euclid's method contribute to the study of perfect numbers?
Euler's sigma function plays a crucial role in summing divisors and proving the Euclid-Euler theorem for even perfect numbers. Euclid's method, on the other hand, involves multiplying primes to generate perfect numbers. Both of these mathematical concepts have significantly contributed to the understanding and exploration of perfect numbers throughout history.
Why is the study of perfect numbers and related mathematical problems important?
Engaging with perfect numbers and other mathematical conundrums offers a unique opportunity to develop problem-solving skills, critical thinking, and a deeper understanding of number theory. The pursuit of such mathematical curiosities not only enriches our knowledge but also lays the foundation for advancements in fields like cryptography and technology, underscoring the enduring value of mathematical exploration and discovery.
Related videos
Vert Dider
Старейшая нерешённая задача [Veritasium]
Veritasium en español
El Problema Sin Resolver Más Antiguo En Matemáticas
Quanta Magazine
The Riemann Hypothesis, Explained
TED
The magic of Fibonacci numbers | Arthur Benjamin | TED
TEDx Talks
Five Principles of Extraordinary Math Teaching | Dan Finkel | TEDxRainier
Summary
00:00
"Oldest Math Problem: Perfect Numbers Mystery"
- The oldest unsolved math problem dates back 2000 years, listed among the most pressing open problems in 2000.
- Mathematicians have used computers to check numbers up to 10^2200 in search of odd perfect numbers.
- Perfect numbers are those where the sum of proper divisors equals the number itself.
- Only 6 and 28 are perfect numbers between 1 and 100, with 496 and 8,128 being the next two.
- Perfect numbers are triangular numbers and the sum of consecutive odd cubes.
- Perfect numbers can be written in binary as consecutive powers of two.
- Euclid's method involves multiplying primes to generate perfect numbers.
- Nicomachus's conjectures about perfect numbers were disproven by later discoveries.
- Mersenne Primes correspond to perfect numbers, with larger numbers being challenging to verify.
- Euler's sigma function sums divisors and proved the Euclid-Euler theorem for even perfect numbers.
13:57
Mathematical Mysteries: Odd Perfect Numbers and Primes
- Euler's sigma function breaks down the sum of prime powers into individual sigmas, revealing that odd prime numbers raised to odd powers result in even numbers, while even powers yield odd numbers.
- Euler's insight led to the conclusion that odd perfect numbers must have exactly one prime raised to an odd power, aligning with Descartes' prediction.
- Mersenne's list of proposed primes aided in the discovery of new perfect numbers, with the 51st Mersenne Prime being 2 to the 82,589,933 minus 1, boasting over 24 million digits.
- The Great Internet Mersenne Prime Search (GIMPS) project allows volunteers to contribute computer power to find Mersenne primes, leading to the discovery of 17 new primes, with a $250,000 prize for the first billion-digit prime.
- The Lenstra and Pomerance Wagstaff conjecture suggests infinitely many Mersenne primes and even perfect numbers, supported by the difficulty in finding these rare primes.
- Spoofs, numbers close to being odd perfect numbers, share properties with odd perfect numbers but have not yet ruled out their existence.
- Heuristic arguments, like Carl Pomerance's, predict the scarcity of perfect numbers, with the belief that odd perfect numbers do not exist.
- The study of number theory, despite lacking real-world applications for centuries, has led to the foundation of modern cryptography, emphasizing the value of pursuing mathematical curiosities.
- Collaboration and persistence in mathematical problem-solving often yield unexpected benefits, as seen in the evolution of non-Euclidean geometries leading to Einstein's general relativity.
- The oldest unsolved problem in math, the existence of odd perfect numbers, continues to intrigue mathematicians, showcasing the enduring pursuit of knowledge for its intrinsic value and potential future applications.
29:00
Exploring Perfect Numbers: A Math Challenge
- Around 10 to 15 people are currently working on the problem of perfect numbers, making it an engaging challenge for high school students interested in mathematics to explore and potentially contribute new insights.
- Engaging with math problems like perfect numbers is valuable because it offers a unique opportunity to develop problem-solving skills and critical thinking, with platforms like Brilliant providing a comprehensive learning experience that combines theoretical knowledge with practical applications in various fields like math, data science, and technology.
