Numbers: Their Tales, Types, and Treasures.
Chapter 8: Special Numbers
8.5.PERFECT NUMBERS
Most mathematics teachers probably told you often enough that everything in mathematics is perfect. While we would then assume that everything in mathematics is truly perfect, might there still be anything more perfect than something else? This brings us to numbers that hold such a title:perfect numbers. This is an official designation by the mathematics community. In the field of number theory, we have an entity called a perfect number, which is defined as a number equal to the sum of its proper divisors (i.e., all the divisors except the number itself).
The smallest perfect number is 6, since 6 = 1 + 2 + 3, which is the sum of all its divisors, excluding the number 6 itself. By the way, 6 is the only number that is both the sum and product of the same three numbers: 6 = 1 × 2 × 3 = 3! Also, . It is also fun to notice that and that both 6 and its square, 36, are triangular numbers (see chapter 4).
The next-larger perfect number is 28 = 1 + 2 + 4 + 7 + 14, which is the sum of all the divisors of 28, excluding 28 itself. The next perfect number is 496, since again, 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248, which is the sum of all of the divisors of 496, excluding 496 itself. The first four perfect numbers were known to the ancient Greeks. They are 6; 28; 496; and 8,128. It was Euclid who came up with a theorem to generalize a procedure to find a perfect number. He said that for an integer, k, if 2^{k} – 1 is a prime number, then one can construct a perfect number using the formula 2^{k}^{–1}(2^{k} – 1). That is, every Mersenne prime (see section 2 of this chapter) gives rise to a perfect number. As noted earlier, 2^{k} – 1 can only be prime when k is a prime number. It should be noted that any perfect number obtained through Euclid's formula is an even perfect number. Leonhard Euler finally proved that every even perfect number can be obtained in this way. It is not known whether there are any odd perfect numbers. None has yet been found.
Using Euclid's method for generating perfect numbers, we get table 8.2, where for the values of k we get 2^{k}^{–1}(2^{k} – 1) as perfect numbers when 2^{k} – 1 is a Mersenne prime number. All presently known Mersenne primes are listed in the appendix, section 3.
k |
Mersenne Prime 2^{k} – 1 |
Perfect Number 2^{k}^{–1}(2^{k} – 1) |
2 |
3 |
6 |
3 |
7 |
28 |
5 |
31 |
496 |
7 |
127 |
8,128 |
13 |
8,191 |
33,550,336 |
17 |
131,071 |
8,589,869,056 |
19 |
524,287 |
137,438,691,328 |
31 |
2,147,483,647 |
2,305,843,008,139,952,128 |
61 |
2,305,843,009,213,693,951 |
2,658,455,991,569,831,744, |
Table 8.2: The first perfect numbers.
As of early 2013, there are forty-eight known Mersenne primes, and, therefore, only forty-eight known perfect numbers. A complete table of these is given in the appendix, section 4. Just to show some of them in their complete form, look at these perfect numbers:
For k = 61: 2^{60}(2^{61} – 1) =
2,658,455,991,569,831,744,654,692,615,953,842,176
For k = 89: 2^{88} (2^{89} – 1) = 191,561,942,608,236,107,294,793,378,
084,303,638,130,997,321,548,169,216
Presently, the largest known perfect number is obtained for k = 57,885,161. This number has 34,850,340 digits.
By observation, we notice some additional properties of perfect numbers. For example, they all seem to end in either a 6 or a 28, and these are preceded by an odd digit. They also appear to be triangular numbers, which are the sums of consecutive natural numbers; for example,
496 = 1 + 2 + 3 + 4 +…+ 30 + 31 = T_{31}.
(See chapter 4 for a definition of the triangular numbers T_{n}). Indeed, if p is a Mersenne prime, then the corresponding perfect number is the triangular number with index p, that is, the sum of the first p integers.
From the work of the Italian mathematician Franciscus Maurolycus (1494–1575) we know that every even perfect number is also a hexagonal number. In general, the nth hexagonal number is given by H_{n} = 2n^{2} – n = n(2n – 1) (see chapter 4, section 7). Inserting n = 2^{k}^{–1}, this formula gives for the 2^{k}^{–1}th hexagonal number the expression 2^{k}^{–1}(2^{k} – 1). We also know that every even perfect number has this form, when 2^{k} – 1 is prime.
To take this a step further, every perfect number after the number 6 is the partial sum of the series:
1^{3} + 3^{3} + 5^{3} + 7^{3} + 9^{3} + 11^{3} +….
For example,
28 = 1^{3} + 3^{3},
496 = 1^{3} + 3^{3} + 5^{3} + 7^{3},
8,128 = 1^{3} + 3^{3} + 5^{3} + 7^{3} + 9^{3} + 11^{3} + 13^{3} + 15^{3}.
This connection between the perfect numbers greater than 6 and the sum of the cubes of consecutive odd numbers is far more than could ever be expected! You might try to find the partial sums for the next few perfect numbers—another challenge for the motivated reader.