Oldest Unsolved Problem In Mathematics
”They may remain forever shrouded in mystery”
Around 100 AD, Nicomachus of Gerasa in the book “Introduction to Arithmetic” presented a classification of numbers based on the concept of perfect numbers. Nicomachus goes on and describes a few results concerning perfect numbers without attempting to prove them.
Two of these result are
- All perfect numbers are even
- There are infinitely many perfect numbers
Both of the statements above haven’t been proved or disproved yet, making them the oldest open problems in mathematics.
“Whether … there are any odd perfect numbers is a most difficult question” — Leonhard Euler
What are perfect numbers?
A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself.
For readability’s sake, let’s annotate the operation of taking the sum of all divisors excluding the number itself with
Using the above annotation, a perfect number is a positive integer n such that
