HARMONIC Numbers are NOT Integers! | Number Theory PROOF!
We explain the intuition and proof behind the beautiful statement that the nth Harmonic number H_n = 1 + 1/2 + 1/3 + ... + 1/n is NEVER an integer unless n = 1. We motivate how the proof is constructed, try out and refine ideas (going on a journey together), and eventually explain the final trick for the proof. The proof involves studying LCMs (least common multiples) in detail. We also reveal the connection between the Harmonic numbers and the most famous open problem in mathematics - the Riemann hypothesis - at the end of the video (the connection is due to the mathematician Jeffrey Lagarias).
The video is designed for anyone interested in abstract mathematics and proof-based mathematics - it is very accessible for students with a strong knowledge of algebra and precalculus, but also of deep interest to advanced math students at any level interested in proof-based math and number-theoretic ideas (e.g., math olympiads and college math).
If you enjoy this video, I would love if you could please like ✅ , subscribe 🎉, and share ⏩ with students, classmates, friends and family - it means so much to help support me in my goal of creating elite infinite free accessible math education worldwide! 😊 If you would like to support my channel beyond this (which of course would mean so much to me 🤗), please check out my "Patreon" page! The contributions here make a world of difference to helping me free up time to produce as much math content as possible (I currently do everything on my own), and there are also exclusive perks available when you signup (e.g., exclusive access to a private Discord server to chat and discuss math with me and other members, insider knowledge on my future YouTube content creation, priority replies to YouTube comments, public acknowledgements as a valued sponsor, a personalized thank you message etc.)! 😊
Become a Patreon Member 🤩: / mathmasterywithamitesh
A HUGE thank you SO much to Alex and Nathan for their continued support of the channel as Gold Members!! 🎊
My Channel Membership 🥇: / @mathmasterywithamitesh
You can also follow me on Instagram! / math_mastery_with_amitesh