Cubes of Perfection

Playing with integers can lead to all sorts of little surprises.

A whole number that is equal to the sum of all its possible divisors—including 1 but not the number itself—is known as a perfect number. For example, the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 equals 6.

Six is the smallest perfect number. The next is 28. Its proper divisors are 1, 2, 4, 7, and 14, and the sum of those divisors is 28.

Incidentally, if the sum works out to be less than the number itself, the number is said to be defective (or deficient). If the sum is greater, the number is said to be abundant. There are far more defective and abundant numbers than perfect numbers. Do abundant numbers actually outnumber defective numbers? I'm not sure.

Steven Kahan, a mathematics instructor at Queens College in Flushing, N.Y., has a long-standing interest in number theory and recreational mathematics. "I often play with number patterns," he says.

In the course of preparing a unit on number theory for one of his classes, he noticed a striking pattern involving the perfect number 28: 28 = 1³ + 3³. He tried the next perfect number: 496 = 1³ + 3³ + 5³ + 7³.

More than 2,000 years ago, the Greek geometer Euclid of Alexandria (325-265 B.C.) proved that if 2^p - 1 is a prime number, then 2^{p - 1}(2^p - 1) is an even perfect number. Primes of the form 2^p - 1 are now known as Mersenne primes, and these numbers figure prominently in the search for the largest known prime.

Leonhard Euler (1707- 1783) proved the converse of Euclid's theorem: All even perfect numbers must have the form specified by Euclid's formula. Hence, every Mersenne prime automatically leads to a new perfect number. There are, at present, 48 known Mersenne primes.

It turns out that a given perfect number 2^{p - 1}(2^p - 1) greater than 6 is expressible as the sum of the cubes of the first n consecutive odd integers, where n = 2^{(p - 1)/2}. For example, if p = 7, then n = 8, and the perfect number 8,128 equals the sum of the cubes of the first eight odd integers: 1³ + 3³ + 5³ + 7³ + 9³ + 11³ + 13³ + 15³. Kahan provides a short proof of this venerable theorem in the April 1998 Mathematics Magazine.

This means that the largest known perfect number, which has 34,850,340 digits, is the sum of the cubes of the first 2^28942580 consecutive odd integers.

Perfect numbers also show other curious patterns. Add the digits of any perfect number greater than 6, then add the digits of the sum together, and so on, until only one digit remains. That final digit is always 1.

28: 2 + 8 = 10; 1 + 0 = 1

496: 4 + 9 + 6 = 19; 1 + 9 = 10; 1 + 0 = 1

8,128: 8 + 1 + 2 + 8 = 19; 1 + 9 = 10; 1 + 0 = 1

Here's another remarkable relationship. The sum of the inverses of the divisors of a perfect number (leaving out 1 but including the number itself) is also 1.

6: 1/2 + 1/3 + 1/6 = 1

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

496: 1/2 + 1/4 + 1/8 + 1/16 + 1/31 + 1/62 + 1/124 + 1/248 + 1/496 = 1

Interesting digit patterns and numerical relationships also arise when Mersenne primes and perfect numbers are written out in binary form. Mersenne primes, for example, consist of unbroken strings of consecutive 1s—57,885,161 of them in the case of the current record holder.

Here are the first four perfect numbers: 110, 11100, 111110000, 1111111000000. See a pattern?

Happy hunting in perfect territory!

Originally posted May 18, 1998

References:

Gullberg, J. 1997. Mathematics: From the Birth of Numbers. W.W. Norton.

Kahan, S. 1998. Perfectly odd cubes. Mathematics Magazine 71(April):131.

Peterson, I. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. Holt.

An introduction to Mersenne primes and perfect numbers can be found at http://www.utm.edu/research/primes/mersenne.shtml.