Perfect numbers

Perfect numbers are [positive] integers

n

for which the sum of divisors equals

2 n

. For example, the divisors of 6 are {1, 2, 3, 6}, and 1 + 2 + 3 + 6 = 12 = 2  ×  6.

The even perfect numbers are of the form

n  =  2 m (2 m  + 1 − 1)  =  2 m σ (2 m ), m ≥ 1,

where

2 m  + 1  −  1

must be prime (called a Mersenne prime, see A000668).

σ (n)  =  σ (2 m ) σ (σ (2 m ))  =  σ (2 m ) σ (2 m  + 1 − 1)  =  σ (2 m ) 2 m  + 1  =  (2 m  + 1 − 1) 2 m  + 1  =  2 m  + 1 (2 m  + 1 − 1)  =  2 n, m ≥ 1,

since powers of 2, with positive exponent, are almost-perfect.

Theorem. (Euclid)

If

2 m  −  1

is a prime number (called a Mersenne prime), then

n = 2 m  − 1 (2 m  −  1)

is a perfect number (see A000396).

Proof. Labelling for our convenience the Mersenne prime as

p = 2 m  −  1

, the divisors of

n

are the powers of 2 from 1 to

2 m  − 1

and each of those powers multiplied by

p

. Since

∑ m  − 1 i  = 0  2 i = 2 m  −  1

, the sum of the divisors of

n

is

(2 m  −  1) + (2 m  −  1) p

. Further rewriting, we get

σ (n) = (2 m  −  1) (1  +  p) = (2 m  −  1) (1  +  2 m  −  1) = 2 m (2 m  −  1) = 2  ×  2 m  − 1 (2 m  −  1) = 2n

, as predicted. □

It wasn’t until Leonhard Euler came along that the converse was proven.

Theorem. (Euler)

If

n

is an even perfect number, it must be the product of a Mersenne prime

p = 2 m  −  1

and the power of two

2 m  − 1

. (See A000079 for the powers of 2.)

Proof. PROOF GOES HERE. □ ^{(Provide proof: PROOF GOES HERE. □)} 

Between Euclid and Euler, medieval mathematicians made some conjectures about perfect numbers that have since been proven false, such as that there is a perfect number between each consecutive power of 10 (there are no perfect numbers between 10000 and 100000, or for that matter between 10000 and 10000000), and that the least significant base 10 digit of each successive perfect number alternates between 6 and 8 (supported by reference only to the first five perfect numbers).

Every even perfect number is a triangular number, since they are a subset of

2 n  − 1 (2 n − 1)  =  (2 n − 1) 2 n 2  =  t 2 n  − 1 ,

where

2 n  −  1

is prime.

Every even perfect number is also an hexagonal number, since they are a subset of

2 n  − 1 (2 n − 1)  =  2 n  − 1 (2 ⋅  2 n  − 1 − 1)  =  h 2 n  − 1 ,

where

2 n  −  1

is prime.

Odd perfect numbers

Main article page: Odd perfect numbers

It is not known whether odd perfect numbers exist or not! Mathematicians have been able to prove all sorts of necessary (but not sufficient) requirements for the existence of such numbers without being able to prove either that they do exist or that they don’t exist.

See also

• Multiply-perfect numbers

• Perfect numbers (singles) (

  σ (n)  −  n = n

  )
• Amicable numbers (pairs) (

  σ (m)  −  m = n

  and

  σ (n)  −  n = m

  )
• Sociable numbers ( 

  k

  -tuples) (

  σ (ni )  −  ni = n(i +1 mod k ) , i = 0 .. k  −  1, k   ≥   3

  )