Старейшая нерешённая задача [Veritasium]
Vert Dider・23 minutes read
The oldest unsolved mathematical problem, known as perfect numbers, has intrigued mathematicians for over 2000 years. Despite progress in finding perfect numbers, the question of whether there are infinitely many of them remains unanswered.
Insights
- Mathematicians have been intrigued by the ancient and beautiful nature of perfect numbers, which are divisible by the sum of their proper divisors, with only a few known examples such as 6, 28, 496, and 8128, each an order of magnitude greater than the previous.
- The search for perfect numbers led to the discovery of Mersenne primes, with efforts like the GIMPS project involving volunteers to find these elusive numbers, hinting at the possibility of an infinite set of Mersenne primes and thus an infinite number of even perfect numbers, highlighting the ongoing mystery surrounding this millennia-old mathematical problem.
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Recent questions
What are perfect numbers?
Numbers divisible by sum of proper divisors.
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Summary
00:00
"Oldest Math Problem Solved After 1600 Years"
- Oldest unsolved mathematical problem, unsolved for 2000 years
- Italian mathematician Jordi named it unsolvable
- Mathematicians checked numbers up to 10^2200 on computers
- Problem still intrigues mathematicians due to its ancient, simple, and beautiful nature
- Definition of perfect numbers: divisible by sum of proper divisors
- Only 6 and 28 perfect among numbers 1-100, 496 and 8128 among 10,000
- Perfect numbers share common traits: each is an order of magnitude greater than the previous, even, and can be written as sums of natural numbers
- Descartes proposed no even perfect numbers beyond Euclid's formula, odd perfect numbers must be a product of prime and square of another number
- Euler found eighth perfect number, introduced Sigma function to sum divisors
- Euler proved even perfect numbers adhere to Euclid's formula, solved problem after 1600 years, confirmed fourth hypothesis of perfect numbers
16:21
Quest for Perfect Numbers: A Mathematical Journey
- Mathematicians continued to search for mysterious perfect numbers, starting with prime numbers from Mars' list.
- Edward proved that 2^67 - 1 is a difficult number, remaining unsolved for 27 years.
- Frank Nelson presented the number 2^67 - 1 as 147 quintillion 573 quadrillion 952 trillion 589 billion 676 million 412 27, receiving applause.
- Raphael Robinson developed a program to find Mersenne primes, discovering five in 10 months.
- By 1952, computers found 12 Mersenne primes, with the largest being 2^2281 - 1.
- In 1996, George Woman initiated a search for Mersenne prime numbers, involving volunteers in the GIMPS project.
- The largest known Mersenne prime by 1994 was 2^859,433 - 1, with 258 digits.
- The hypothesis suggests an infinite set of Mersenne primes, indicating infinitely many even perfect numbers, but the question of the number of perfect numbers remains unsolved.
