628 Perfect numbers
24.1 Perfect numbers .................
A number is perfect is the sum of its proper divisors (including 1) is
equal to the number itself.
Theorem 24.1(Euclid).If1+2+2^2 +···+2k−^1 =2k− 1 is a prime
number, then 2 k−^1 (2k−1)is a perfect number.
Note: 2 k− 1 is usually called thek-th Mersenne number and denoted
byMk.IfMkis prime, thenkmust be prime.
Theorem 24.2(Euler).Every even perfect number is of the form given
by Euclid.
Open problem
Does there exist anoddperfect number?
Theorem-joke 24.1(Hendrik Lenstra).Perfect squares do not exist.^1
Proof.Supposenis a perfect square. Look at the odd divisors ofn.
They all divide the largest of them, which is itself a square, sayd^2. This
shows that the odd divisors ofncome in pairsa,bwherea·b=d^2. Only
dis paired to itself. Therefore the number of odd divisors ofnis also
odd. In particular, it is not 2 n. Hencenis not perfect, a contradiction:
perfect squares don’t exist.
(^1) Math. Intelligencer, 13 (1991) 40.