Oldest Unsolved Problem in Math

Overview

• Dates back 2000 years.
• Notable mathematicians have attempted to solve it, but failed.
• Listed by Pier Giorgio Odofredi as one of four pressing open problems in 2000.
• Solving it may involve finding a single number.
• Computers have checked numbers up to 10^2200 without success.

The Problem

• Key Question: Do any odd perfect numbers exist?

What is a Perfect Number?

• A perfect number equals the sum of its proper divisors.
  • Example: 6 (Divisors: 1, 2, 3; Sum = 1 + 2 + 3 = 6).
• Not Perfect: 10 (Divisors: 1, 2, 5; Sum = 1 + 2 + 5 = 8).
• Only perfect numbers between 1 and 100: 6 and 28.
• Next two up to 10,000: 496 and 8128.
• Ancient Greeks knew only these numbers for over a thousand years.

Characteristics of Perfect Numbers

• Each subsequent perfect number is one digit longer than its predecessor.
• The last digits alternate between 6 and 8.
• All known perfect numbers are even.
• Can be expressed as sums of consecutive numbers or cubes:
  • 6 = 1 + 2 + 3
  • 28 = 1 + 2 + 3 + 4 + 5 + 6 + 7
• Binary representation shows a pattern of 1s followed by 0s.

Euclid's Contribution

• Euclid's Discovery (300 BC):
  • Created a method to generate even perfect numbers:
    • Start with 1, double it repeatedly, and sum those results.
    • If the sum plus one is prime, multiply by the last number to get a perfect number.
• Formula: For prime p, perfect number = (2^(p-1))(2^p - 1).

Nicomachus' Conjectures (13th Century)

1. The nth perfect number has n digits.
2. All perfect numbers are even.
3. All perfect numbers end in 6 and 8 alternately.
4. Euclid's algorithm produces every even perfect number.
5. There are infinitely many perfect numbers.

Disproving Some Conjectures

• 5th perfect number disproves first conjecture (not 8 digits long).
• Subsequent discoveries disproved the third conjecture about alternating digits.

Renaissance and Mersenne Primes

• French polymath Marin Mersenne studied numbers of the form 2^p - 1 to find primes, leading to perfect numbers.
• Mersenne primes correspond to perfect numbers discovered so far.
• Descartes suggested that if odd perfect numbers exist, they must be of a specific form.

Euler's Work

• Leonhard Euler (18th Century):
  • Discovered the eighth perfect number.
  • Proved all even perfect numbers follow Euclid’s form, confirming Nicomachus' fourth conjecture.
  • Suggested odd perfect numbers must have a particular structure: one odd prime raised to an odd power, all others even.

Modern Developments

• Ongoing Research: No proof on existence of odd perfect numbers yet.
• Recent algorithms have set bounds: must be larger than 10^2200.
• Spoofs: Numbers close to odd perfect numbers that fail to meet all criteria.
• Conditions add complexity to the search for odd perfect numbers.

Current Status

• Only 51 Mersenne primes discovered despite rapid computer advancements.
• The Leinster-Palmerantz-Wagstaff conjecture suggests infinitely many Mersenne primes exist.
• The search for odd perfect numbers remains the oldest unsolved problem in mathematics.

Significance of the Problem

• Many mathematicians still study it despite no direct applications.
• Historically, issues in number theory have led to unexpected real-world applications, notably in cryptography.
• Curiosity-driven mathematical exploration can yield innovative solutions.

Conclusion

• The problem of odd perfect numbers remains open, with mathematicians continuing to explore conditions that might limit their existence.
• Heuristic arguments suggest the improbability of their existence, but no conclusive proof yet.