The Oldest Unsolved Problem in Math

Step-by-Step Tutorial:

1. Understanding the Problem:

• The oldest unsolved problem in math is about odd perfect numbers.
• Perfect numbers are numbers where the sum of their proper divisors equals the number itself.
• The video discusses the history of perfect numbers, their patterns, and the search for odd perfect numbers.

2. Definition of Perfect Numbers:

• Perfect numbers are numbers where the sum of their proper divisors equals the number itself.
• For example, 6 is a perfect number because 1 + 2 + 3 = 6.

3. Finding Perfect Numbers:

• Most numbers between 1 and 100 either overshoot or undershoot the definition of a perfect number.
• The only perfect numbers known to ancient Greeks were 6 and 28.
• By following a specific pattern, mathematicians have discovered more perfect numbers like 496 and 8,128.

4. Euclid's Contribution:

• Euclid discovered a pattern for generating perfect numbers using prime numbers.
• He found a formula that generates even perfect numbers: (2^{(p-1)} \times (2^p - 1)), where (p) is a prime number.

5. Mersenne Primes and Perfect Numbers:

• Marin Mersenne studied numbers of the form (2^p - 1) to find Mersenne Primes.
• Mersenne Primes correspond to perfect numbers.
• The search for Mersenne Primes led to the discovery of new perfect numbers.

6. Modern Discoveries:

• In 1952, Raphael Robinson wrote a computer program to find Mersenne Primes.
• The Great Internet Mersenne Prime Search (GIMPS) project involves volunteers searching for Mersenne Primes.
• Modern computers have helped discover large Mersenne Primes with millions of digits.

7. Unsolved Problems:

• The search for odd perfect numbers continues.
• Spoofs are numbers close to being odd perfect numbers but fall short.
• The Lenstra and Pomerance Wagstaff conjecture predicts the existence of infinitely many Mersenne Primes and perfect numbers.

8. Impact of the Problem:

• While the search for perfect numbers may seem abstract, the history of mathematics shows how curiosity-driven research can lead to practical applications.
• Number theory has influenced cryptography and other real-world applications.

9. Learning Opportunities:

• Platforms like Brilliant offer opportunities to explore math, data science, programming, and technology.
• By following your curiosity and building skills, you can contribute to solving complex problems like the search for perfect numbers.

10. Conclusion:

• The search for odd perfect numbers remains an ongoing challenge in mathematics.
• Exploring this problem not only advances mathematical knowledge but can also lead to unexpected practical applications.

By following the steps outlined above, you can gain a deeper understanding of the oldest unsolved problem in math and its significance in the field of mathematics.