The Oldest Unsolved Problem in Math

Do odd perfect numbers exist? Head to to start your free 30-day trial, and the first 200 people get 20% off an annual premium subscription. If you’re looking for a molecular modeling kit, try Snatoms, a kit I invented where the atoms snap together magnetically - 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: These sources were particularly helpful: Perfect numbers via MacTutor - Cai, T. (2022). Perfect numbers and fibonacci sequences. World Scientific. - Dunham, W. (2022). Euler: The master of us all (Vol. 22). American Mathematical Society. - ▀▀▀ References: 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. - Perfect number via Wikipedia - Introduction to Arithmetic via HalthiTrust - Nicomachus of Gerasa via MacTutor - 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. - Ibn Fallus via Wikipedia - Mersenne prime via Wikipedia - List of Known Mersenne Prime Numbers - Marin Mersenne via MacTutor - Leonhard Euler via Wikipedia - Frank Nelson Cole via Wikipedia - GIMPS History via - EFF Cooperative Computing Awards via EFF - Jonathan Pace via Primewiki - Book with just one number sells out in Japan via BastillePost - Predicted distribution of Mersenne primes via John D. Cook - Euler’s Odd Perfect Numbers Theorem via Cantor’s Paradise - A Perfect (Math) Mystery via Medium - 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. - Ochem, P., & Rao, M. (2012). Odd perfect numbers are greater than 10¹⁵⁰⁰. Mathematics of Computation, 81(279), 1869-1877. - Mathematicians Open a New Front on an Ancient Number Problem via Quantamagazine - Descartes number via Wikipedia - 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. - Pomerance’s Heuristic that Odd Perfect Numbers are Unlikely via - Images & Video: Clip of Piergiorgio Odifreddi - Euclid’s Elements 1 via Claymath - Euclid’s Elements 2 via Claymath - Euclid’s Elements 3 via Claymath - Diophanti - Gauss book - Euler’s Archive 1 - Euler’s Archive 2 - ▀▀▀ 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