Perfect Numbers
The Greeks were enchanted by numbers that equal the sum of their own proper divisors —
numbers in perfect balance with their parts. 6 = 1 + 2 + 3 is the
first. These perfect numbers are rare, beautiful, and hide one of the oldest
unsolved problems in mathematics.
Definition
A positive integer n is perfect if its proper divisors
(all divisors except n itself) sum to n —
equivalently, using the
divisor sum
\sigma that includes n:
\sigma(n) = 2n.
Check 28: its divisors 1,2,4,7,14,28 sum to
56 = 2 \cdot 28. Numbers with \sigma(n) < 2n
are called deficient, and those with \sigma(n) > 2n
abundant.
The Euclid–Euler theorem
Euclid noticed that perfect numbers are forged from
Mersenne primes;
two thousand years later Euler proved that every even perfect number arises this way.
An even number is perfect if and only if it has the form
n = 2^{p-1}\left(2^{p} - 1\right), \qquad \text{where } 2^{p} - 1 \text{ is prime.}
With p = 2: 2^{1}(2^{2}-1) = 2 \cdot 3 = 6.
With p = 3: 4 \cdot 7 = 28. Each new Mersenne
prime delivers exactly one new even perfect number.
Two ancient mysteries
Because perfect numbers ride on Mersenne primes, we do not know whether there are
infinitely many — it is open precisely because the infinitude of Mersenne primes is open.
And no one has ever found an odd perfect number, nor proved that none exists. Both
questions have stood unanswered for over two millennia — a humbling reminder of how deep the
simplest definitions can run.