Perfect Numbers (VIDEO!)
The numbers 6, 28, 496 and 8128, for example, are “perfect” in the sense that if you sum of all their factors (except for the whole number itself) you get the original number. For example, 28 has factors 1, 2, 4, 7, 14 and 28 and, ignoring the last one, 1+2+4+7+14 does equal 28!
The Greeks noticed that each of these numbers is triangular: 6 is the 3rd triangle number, 28 is the 7th triangle number, 496 is the 31st triangle number, 8128 is the 127th triangle number, and so on, and the numbers 3, 7, 31, 127 are each primes one less than a power of two (“Mersenne Primes”)!
It wasn’t until some 2000 years later that Euler was able to prove that every (even) perfect number must have this form. We prove it too in this video.
(WARNING: My technique this time is a bit fast paced and a bit messy, but all the details are present and accounted for in the video!)