Lecture Notes: The Oldest Unsolved Problem in Mathematics

Introduction

• The problem of odd perfect numbers is a 2000-year-old unsolved problem in mathematics.
• Despite significant effort from mathematicians, the existence of odd perfect numbers remains unknown.
• In 2000, Pier Giorgio Odifreddi included this among four major unsolved problems in math.

Perfect Numbers

• A perfect number is a positive integer equal to the sum of its proper divisors (excluding itself).
  • Example: 6, where the divisors 1, 2, and 3 sum to 6.
• Known perfect numbers up to 10,000: 6, 28, 496, 8,128.
• Characteristics:
  • Each subsequent perfect number is one digit longer.
  • Alternating end digits (6 and 8) in even perfect numbers.

Triangular and Binary Patterns

• Perfect numbers are also triangular numbers and can be expressed as sums of consecutive odd cubes.
• In binary form, perfect numbers appear as strings of ones followed by zeros (powers of two).

Euclid's Contribution

• Euclid discovered a method for generating even perfect numbers:
  • Multiply a sum of powers of two up to 2^(p-1) by 2^(p-1), if the sum is prime.
  • This results in even perfect numbers.

Historical Conjectures

• Nicomachus’ five conjectures about perfect numbers:
  1. The nth perfect number has n digits.
  2. All perfect numbers are even.
  3. Perfect numbers end in 6 and 8 alternately.
  4. Euclid's method produces every even perfect number.
  5. Infinitely many perfect numbers exist.
• Conjectures 1 and 3 were disproved in later centuries.

Developments by Euler

• Euler proved that every even perfect number has Euclid’s form, confirming Nicomachus' fourth conjecture.
• Developed the sigma function, a powerful tool in number theory.
• Attempted to tackle odd perfect numbers but was unable to fully solve the problem.

Mersenne Primes

• Discovered by Marin Mersenne, these primes are of the form 2^p - 1.
• Mersenne numbers are crucial to finding perfect numbers.
• Mersenne primes have been discovered extensively with the aid of computers in modern times (e.g., GIMPS project).

Current Understanding and Efforts

• No odd perfect numbers have been found, and they must be exceedingly large if they exist.
• Various conditions have been proposed that odd perfect numbers would need to meet, forming a 'web of conditions'.
• Spoofs are numbers close to being odd perfect but fail upon closer inspection.
• Heuristic arguments suggest odd perfect numbers are unlikely to exist, although they are not proofs.

Conclusion

• The problem of odd perfect numbers remains unsolved and continues to intrigue mathematicians.
• Despite the lack of real-world applications, studying such problems is valuable for the development of mathematical thought and techniques.
• The pursuit of this and other mathematical problems serves as a testament to human curiosity and intellectual effort.

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Additional Information

• The lecture ends with a discussion on the potential long-term benefits of pursuing seemingly abstract mathematical problems.
• Sponsor: Brilliant.org, a platform for learning through interactive problem-solving.