The Oldest Unsolved Problem in Math

Summary by Krater

This video explores the history and mathematics of perfect numbers, starting with Euclid and following the centuries-long search for even and odd perfect numbers.

From the video

Answers: Do any odd perfect numbers exist?

• perfect numbers
• Mersenne primes
• sigma function
• number theory
• odd perfect numbers

What it concludes

• Nicomachus's first and third conjectures that the nth perfect number has n digits and that all perfect numbers end in 6 and 8 alternately are false.
• Euler proved that every even perfect number has Euclid's form and discovered the 8th perfect number.
• Euler proved that an odd perfect number must satisfy N = p^(4k+1) * M^2.
• No odd perfect numbers have been discovered.
• Pomerance's heuristic predicts that between 10^2200 and infinity there are no more than 10^-540 perfect numbers of the form N = p * M^2.