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? 😉
| 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.
| 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).
| 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 × (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).
| 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
| 1 | + | 1 | + | 1 | + | 1 | = | 2 |
| 1 | 2 | 3 | 6 |
| 1 | + | 1 | + | 1 | + | 1 | + | 1 | + | 1 | = | 2 |
| 1 | 2 | 4 | 7 | 14 | 28 |
| 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.
| 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.
| 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 😉
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