Do odd perfect numbers exist? Head to https://brilliant.org/veritasium to start your free 30-day trial, and the first 200 people get 20% off an annual premium subscription.

Special thanks to our Patreon supporters! Join this list to help us keep our videos free, forever:
https://ve42.co/PatreonDEB

A massive thank you to Prof. Pace Nielsen for all his time and help with this video.

A big thank you to Dr. Asaf Karagila, Pascal Ochem, Prof. Tianxin Cai, and Prof. William Dunham for their expertise and help.

To try GIMPS out yourself: https://ve42.co/GIMPS

These sources were particularly helpful:
Perfect numbers via MacTutor - https://ve42.co/MTPerfect
Cai, T. (2022). Perfect numbers and fibonacci sequences. World Scientific. - https://ve42.co/Cai2022
Dunham, W. (2022). Euler: The master of us all (Vol. 22). American Mathematical Society. - https://ve42.co/Dunham2022

▀▀▀
References:
   • Perfect Numbers and Mersenne Primes -...  
   • Perfect Number Proof - Numberphile  
Dickson, L. E. (1919). History of the Theory of Numbers.. (Vol. 1). Carnegie Institution of Washington.
Knill, O. (2007). The oldest open problem in mathematics. NEU Math Circle, December2. - https://ve42.co/Knill2007
Perfect number via Wikipedia - https://ve42.co/WikiPerfect
Introduction to Arithmetic via HalthiTrust - https://ve42.co/IntroArithmetic
Nicomachus of Gerasa via MacTutor - https://ve42.co/MTNicomachus
Sonja, B. (1988). The First Perfect Numbers and Three Types of Amicable Numbers in a Manuscript on Elementary Number Theory by Ibn Fellûs. Erdem, c. IV, 11. - https://ve42.co/Sonja1988
Ibn Fallus via Wikipedia - https://ve42.co/WikiFallus
Mersenne prime via Wikipedia - https://ve42.co/WikiMP
List of Known Mersenne Prime Numbers - https://ve42.co/ListOfMP
Marin Mersenne via MacTutor - https://ve42.co/MTMersenne
Leonhard Euler via Wikipedia - https://ve42.co/WikiEuler
Frank Nelson Cole via Wikipedia - https://ve42.co/WikiFNCole
GIMPS History via Mersenne.org - https://ve42.co/GIMPSHistory
EFF Cooperative Computing Awards via EFF - https://ve42.co/EFFAwards
Jonathan Pace via Primewiki - https://ve42.co/PWikiPace
Book with just one number sells out in Japan via BastillePost - https://ve42.co/PrimeBook
Predicted distribution of Mersenne primes via John D. Cook - https://ve42.co/JDCookMP
Euler’s Odd Perfect Numbers Theorem via Cantor's Paradise - https://ve42.co/EulerOPN
A Perfect (Math) Mystery via Medium - https://ve42.co/Machado2024
Brent, R. P., Cohen, G. L., & te Riele, H. J. (1991). Improved techniques for lower bounds for odd perfect numbers. Mathematics of Computation, 57(196), 857-868. - https://ve42.co/Brent1991
Ochem, P., & Rao, M. (2012). Odd perfect numbers are greater than 10¹⁵⁰⁰. Mathematics of Computation, 81(279), 1869-1877. - https://ve42.co/Ochem2012
Mathematicians Open a New Front on an Ancient Number Problem via Quantamagazine - https://ve42.co/QuantaSpoofs
Descartes number via Wikipedia - https://ve42.co/WikiDescartesNumber
Andersen, N., Durham, S., Griffin, M. J., Hales, J., Jenkins, P., Keck, R., ... & Wu, D. (2022). Odd, spoof perfect factorizations. Journal of Number Theory, 234, 31-47. - https://ve42.co/Andersen2022
Pomerance’s Heuristic that Odd Perfect Numbers are Unlikely via OddPerfect.org - https://ve42.co/Heuristic

Images & Video:
Clip of Piergiorgio Odifreddi -    • Odifreddi da Gramellini: piccola lezi...  
Euclid’s Elements 1 via Claymath - https://ve42.co/CM1
Euclid’s Elements 2 via Claymath - https://ve42.co/CM2
Euclid’s Elements 3 via Claymath - https://ve42.co/CM3
Diophanti - https://ve42.co/Diophanti
Gauss book - https://ve42.co/GaussDis
Euler’s Archive 1 - https://ve42.co/Euler1
Euler’s Archive 2 - https://ve42.co/Euler2

▀▀▀
Special thanks to our Patreon supporters:
Anton Ragin, Balkrishna Heroor, Bertrand Serlet, Bill Linder, Blake Byers, Burt Humburg, Chris Harper, Dave Kircher, David Johnston, Diffbot, Evgeny Skvortsov, Garrett Mueller, Gnare, I.H., John H. Austin, Jr., john kiehl, Josh Hibschman, Juan Benet, KeyWestr, Lee Redden, Marinus Kuivenhoven, Matthias Wrobel, Max Paladino, Meekay, meg noah, Michael Krugman, Orlando Bassotto, Paul Peijzel, Richard Sundvall, Sam Lutfi, Stephen Wilcox, Tj Steyn, TTST, Ubiquity Ventures, wolfee

▀▀▀
Directed by Casper Mebius
Written by Casper Mebius and Derek Muller
Edited by Peter Nelson
Illustrated by Jakub Misiek
Animated by Fabio Albertelli, Ivy Tello, David Szakaly, Alondra Vitae, Alex Drakoulis, and Leigh Williamson
Filmed by Derek Muller, Raquel Nuno, and Peter Nelson
Additional research by Aaron Santos, Camilla Machado, and Gregor Čavlović
Produced by Casper Mebius, Gregor Čavlović, Han Evans, and Derek Muller

Thumbnail by Ren Hurley
Additional video/photos supplied by Getty Images and Pond5
Music from Epidemic Sound