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: https://www.youtube.com/watch?v=T0xKHwQH-4I&pp=ygUPcGVyZmVjdCBudW1iZXJz https://www.youtube.com/watch?v=q8n15q1v4Xo&pp=ygUPcGVyZmVjdCBudW1iZXJz 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 - https://www.youtube.com/watch?v=y0paHEA8-gA 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