# The Beauty of Perfect Numbers

Jun 09 2011

Have you heard of perfect numbers? Six is the first perfect number. It is regarded as “perfect” because it is equal to the sum of its proper divisors (or aliquot sum); these divisors include 1 but exclude the number itself. Thus, the proper divisors of 6 are 1, 2 and 3, and their sum 1 + 2 + 3 = 6. Perfect, don’t you think? ðŸ˜‰

Here are the first 6 perfect numbers:

Perfect Number Proper Divisors (arranged in column pairs)
6   1 2
3
28   1 2 4
14 7
496   1 2 4 8 16
248 124 62 31
8128   1 2 4 8 16 32 64
4064 2032 1016 508 254 127
33550336   1 2 4 8 16 32 64 128 256 512 1024 2048 4096
16775168 8387584 4193792 2096896 1048448 524224 262112 131056 65528 32764 16382 8191
8589869056 Too many to list…

To date, 47 perfect numbers have been found; however, from the 40th to 47th perfect numbers, it has not been verified if other perfect numbers exist between them. And it is not known if there are infinitely more perfect numbers beyond, which is one of the unsolved mathematical problems of today.

Incidentally, none of the perfect numbers found are odd. No proof exists that there isn’t an odd perfect number — this is yet another maths problem that remains unsolved. Even perfect numbers are linked to the following topic…

#### Mersenne Primes and Perfect Numbers

Marin Mersenne (1588-1648), a freelance French mathematician, found that 2p – 1 is prime for some values of p, which itself must also be prime. Prime numbers of this form are called Mersenne primes.

Let’s take a look at the first 7 prime numbers and see if they produce Mersenne primes:

p 2p – 1 Mersenne Prime?
2 22 – 1 = 3 Yes
3 23 – 1 = 7 Yes
5 25 – 1 = 31 Yes
7 27 – 1 = 127 Yes
11 211 – 1 = 2047 = 23 x 89 No
13 213 – 1 = 8191 Yes
17 217 – 1 = 131071 Yes

Mersenne primes are very rare. There are 78,498 prime numbers below a million, but only 33 of these (which p assumes) produce primes in the form 2p – 1.

The quest of discovering more perfect numbers saw mathematicians use rather “magical” formulas. Indeed, in the third century BCE, Euclid proved that 2p-1 x (2p – 1) is a perfect number when 2p – 1 is prime, or more correctly a Mersenne prime as we have just learned. Interestingly, Leonhard Euler proved in 1770, some 2000 years later, that every even perfect number is of the form 2p-1 x (2p – 1).

Therefore, Mersenne primes and perfect numbers are tightly linked. The number of (even) perfect numbers found so far is 47, because that’s the number of Mersenne primes known today. The list below illustrates this relationship, referencing the first 6 perfect numbers we saw earlier:

S/No p Mersenne Prime 2p-1 x (2p – 1) Perfect Number
1 2 3 21 x 3 6
2 3 7 22 x 7 28
3 5 31 24 x 31 496
4 7 127 26 x 127 8128
5 13 8191 212 x 8191 33550336
6 17 131071 216 x 131071 8589869056

The list of Mersenne primes goes on till the 47th prime, which produces the 25,956,377-digit perfect number 2243,112,608 Ã— (2243,112,609 – 1). Wow!

#### Sum of Natural Numbers, or Arithmetic Series

A perfect number can also be expressed as a sum of natural numbers (or positive integers) up to a specific number, i.e. an arithmetic series. The first term of the series is 1 and the last term is defined by 2p – 1, where p is related to the Mersenne prime that forms that perfect number (as presented earlier).

Consider the following perfect numbers and corresponding arithmetic series:

 6 = 1 + 2 + 3 = 22-1 x (22 – 1) 28 = 1 + 2 + 3 + … + 7 = 23-1 x (23 – 1) 496 = 1 + 2 + 3 + … + 31 = 25-1 x (25 – 1) 8128 = 1 + 2 + 3 + … + 127 = 27-1 x (27 – 1) 33550336 = 1 + 2 + 3 + … + 8191 = 213-1 x (213 – 1) 8589869056 = 1 + 2 + 3 + … + 131071 = 217-1 x (217 – 1) \ s ame /

#### Sum of Odd Cubes

Except for the first perfect number, 6, all even perfect numbers can be expressed as the sum of cubes of successive odd numbers, starting from 1. The number of terms in the summation is determined by 2(p-1)/2. A few examples are shown below:

 28 = 13 + 33 2(3-1)/2 = 2 terms 496 = 13 + 33 + 53 + 73 2(5-1)/2 = 4 terms 8128 = 13 + 33 + 53 + … + 153 2(7-1)/2 = 8 terms 33550336 = 13 + 33 + 53 + … + 1273 2(13-1)/2 = 64 terms 8589869056 = 13 + 33 + 53 + … + 5113 2(17-1)/2 = 256 terms

#### Sum of Reciprocals of Divisors

Interestingly, when you add up all the reciprocals of the divisors of a perfect number, you will always get 2 as the answer. For example, 6 has the divisors 1, 2, 3 and 6. The sum of the reciprocals of divisors is:

 1 + 1 + 1 + 1 = 2 1 2 3 6

Similarly, for 28, we’d get:

 1 + 1 + 1 + 1 + 1 + 1 = 2 1 2 4 7 14 28

In general, if a perfect number P has n divisors d1 to dn, where d1 = 1 and dn = P = d1 + d2 + d3 + … + dn-1 (aliquot sum), then

 1 + [ 1 + 1 + 1 + … + 1 ] = 1 + dn-1 + dn-2 + dn-3 + … + d1 = 1 + 1 = 2 d1 d2 d3 d4 dn dn

#### Digital Root

The digital root of a number is obtained by summing the digits of that number and repeatedly summing the digits of ensuing sums until a single digit is reached. It is also called the repeated digital sum. For example, the digital root of 12345 is 6, because 1 + 2 + 3 + 4 + 5 = 15, and 1 + 5 = 6.

For even perfect numbers other than 6, their digital roots are always 1. Cool! Let’s try a few examples:

 28 gives 2 + 8 = 10; 1 + 0 = 1 496 gives 4 + 9 + 6 = 19; 1 + 9 = 10; ditto 8128 gives 8 + 1 + 2 + 8 = 19; 1 + 9 = 10; ditto 33550336 gives 3 + 3 + 5 + 5 + 0 + 3 + 3 + 6 = 28; 2 + 8 = 10; ditto 8589869056 gives 8 + 5 + 8 + 9 + 8 + 6 + 9 + 0 + 5 + 6 = 64; 6 + 4 = 10; ditto

The first perfect number 6 is the exception here because it’s p value (which is 2) is even, in the formula 2p-1 x (2p – 1). All the other (even) perfect numbers have p values that are odd (and prime, of course; recall how Mersenne primes relate to perfect numbers).

#### Binary Representation

To write a (decimal) perfect number in binary, simply write p ones followed by (p – 1) zeros. This is so straightforward because a perfect number is actually the product of 2 numbers: 2p-1 and 2p – 1. Each factor in this product has a unique property in the binary system, as follows:

• 2p – 1
Decrementing a binary number which is an exact (positive) power of 2 always results in a string of ones; the number of ones in this string is the same as that binary power. So 2p – 1 always gives a string of p ones.
• 2p-1
Multiplying a number by the nth power of 2 (n>0) means shifting that number, represented in binary, left by n places, or effectively padding the number with as many zeros as places shifted. Therefore, multiplying by 2p-1 means adding (p – 1) trailing zeros.
Thus, the first 6 perfect numbers are looking pretty in binary:

 6 = 1102 28 = 111002 496 = 1111100002 8128 = 11111110000002 33550336 = 11111111111110000000000002 8589869056 = 1111111111111111100000000000000002

#### Not So Perfect?

I knew about perfect numbers and Mersenne primes for a while now, but was pleasantly surprised by the other “cute” properties of perfect numbers — arithmetic series, sum of odd cubes, sum of reciprocals of divisors, etc.

It is true that we can learn new things each day, so why not adopt our WittyCulus attitude: keeping our minds open and our eyes keen. I am no Maths genius, but I continue to be fascinated by this wonderful subject which I like. Go on, further your own learning in any chosen field — it’s never too late ðŸ˜‰