What are perfect numbers? We'll go over the few equivalent definitions of perfect numbers, examples and non-examples, discuss some history and conjectures, and introduce the Euclid-Euler theorem, which connects perfect numbers to the Mersenne primes! #NumberTheory
Perfect numbers are positive integers equal to half the sum of their divisors. The divisors of 6, for example, are 1, 2, 3, and 6, and those divisors add to 12. Since 6 is half of 12, the sum of its divisors, it is perfect. Equivalently, we could consider only the proper divisors, which exclude the number itself. The sum of the proper divisors of a numbers is called the aliquot sum. So for 6, the aliquot sum s(6) = 1+2+3 = 6. Since it equals its aliquot sum, it is perfect!
Mersenne Primes: (coming soon)
Proof of the Euclid-Euler Theorem: (coming soon)
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