Oldest Unsolved Problem in Mathematics

Overview

• Problem dates back 2000 years.
• Attempted by many famous mathematicians.
• Listed by Piergiorgio Odifreddi in 2000 among four pressing open problems.
• Could be solved by finding a single number.
• Computers checked numbers up to 10^2200 without success.

Why It Captivates Mathematicians

• Its age, simplicity, and beauty make it intriguing.

The Problem: Odd Perfect Numbers

• Definition of Perfect Numbers:
  • Example: 6 is perfect because its proper divisors (1, 2, 3) sum to 6.
  • 10 is not perfect because its proper divisors (1, 2, 5) sum to 8.
• Known perfect numbers:
  • Up to 10,000: 6, 28, 496, 8128.
  • Ancient Greeks only knew 6 and 28.

Patterns in Perfect Numbers

• Perfect numbers increase in digit length.
• They alternate ending digits between 6 and 8.
• All even perfect numbers can be expressed as sums of consecutive numbers and odd cubes.
• In binary, perfect numbers appear as strings of ones followed by zeros.

Euclid's Contribution (300 BC)

• Discovered a formula for generating even perfect numbers:
  • Formula: 2^(p-1) * (2^p - 1) where (2^p - 1) is prime.
  • Examples: 6 = 2^1 * (2^2 - 1), 28 = 2^2 * (2^3 - 1), etc.*

Nicomachus's Conjectures (13th Century)

• Conjectures about perfect numbers:
  1. The n-th perfect number has n digits.
  2. All perfect numbers are even.
  3. Perfect numbers alternate ending in 6 and 8.
  4. Euclid's algorithm produces all even perfect numbers.
  5. Infinitely many perfect numbers exist.
• Some conjectures later disproven by Ibn Fallus in the 13th century.

Development Through the Ages

• Renaissance Europe rediscovered perfect numbers.
• Marin Mersenne studied primes leading to Mersenne primes.
• Descartes speculated about odd perfect numbers.
• Euler made breakthroughs, including the 8th perfect number.

Euler's Breakthroughs

1. Discovered the 8th perfect number.
2. Proved every even perfect number has Euclid's form (Euclid-Euler theorem).
3. Investigated properties of odd perfect numbers.

Ongoing Research

• No new perfect numbers were discovered for 150 years.
• Doubts about odd perfect numbers exist; many conjectures suggest they might not exist.
• Modern computations and research (GIMPS) continue searching Mersenne primes.

Current Status

• Over 51 Mersenne primes found, suggesting infinitely many even perfect numbers.
• Odd perfect numbers remain unproven, with researchers estimating they must be larger than 10^2200.
• Researchers are now focusing on properties of "spoofs" (close to odd perfect numbers).

Future Directions

• Heuristic arguments suggest limited existence of odd perfect numbers.
• The problem remains a significant area of number theory research, driven by mathematicians' curiosity.

Conclusion

• The pursuit of solving odd perfect numbers may not lead to practical applications but fosters mathematical development and exploration.