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.

Beyond Ultra Large and Extra Small

Metric (SI) prefixes come in handy for expressing both large and small quantities. So we have gigawatt power systems and terabyte hard drives at the high end, and we have nanoscale electronics and femtosecond lasers at the low end.

Advances in scientific research and technology keep pushing us to new frontiers in the domain of metric prefixes.

In a 1998 article in Scientific American, Philip Morrison and Phylis Morrison described the escalation in the amount of stored knowledge since the days of the famed library in ancient Alexandria. The library probably shelved the equivalent of 50,000 books today. Counting the literature of China, India, Iraq, and Iran, the total number of distinct books available 2,000 years ago might have been about 100,000.

In 1945, a librarian at Harvard estimated that "worthwhile" printed material amounted to about 10 million books. With one byte as a unit of information representing roughly the equivalent of a single character of English text, it's possible to store that text as 10 trillion bytes, or 10 terabytes, of data. Images, drawings, photographs, audio and video recordings, and huge databases, however, require much larger amounts of storage space.

The U.S. Library of Congress now holds more than 34.5 million books and other print materials. Because it also has 3.3 million sound recordings, 13.4 million photographs, 5.4 million maps, and other byte-hungry material, however, the total store comes to a couple of petabytes (a million billion bytes).

One can plausibly estimate that fresh text, including newspapers, amounts to less than 100 terabytes annually. Recorded music, films, photographs (including family snapshots), and other materials add considerably to that accumulation.

"The biggest byte makers are the television stations of the world," the Morrisons wrote. "Although it is hard to correct for innumerable repeats, our best source puts their originality at one tenth of all they send out and so allots them under 100 petabytes annually."

To assess the totality of information production, one can also try to include ephemeral signals conveyed from one person to another. The sounds of telephone calls add up to some 1,000 petabytes worldwide, or a few exabytes. Face-to-face speech provides several more exabytes of data.

At this stage, the summing gets trickier. The Morrisons simply concluded, "No estimate of the eventual human store seems quite credible as yet."

In a 2011 online article in Science Express, Martin Hilbert and Priscila Lopez provided new estimates of the world’s technological capacity to store, communicate, and compute information. They estimated that in 2007 humankind was able to store 2.9 x 10²⁰ optimally compressed bytes, communicate nearly 2 x 10²¹ bytes, and carry out 6.4 x 10¹⁸ instructions per second on general-purpose computers.

These estimates put us in a realm of metric prefixes that relatively few people yet know. In 1991 as part of the International System of Units (SI), the General Conference on Weights and Measures (Conférence Générale des Poids et Mesures) adopted new prefixes representing 10²¹, 10²⁴, 10^-21, and 10^-24. This means that 1,000 exabytes equals 1 zettabyte (ZB), and 1,000 zettabytes equals 1 yottabyte (YB).

I first encountered the new prefixes in 1993 when researchers measured voltages in a superconducting circuit so small that they had to use the term milliattovolt, where "atto" stands for 10^-18. The proper term is zeptovolt.

Here's the current table of official metric (SI) prefixes:

Power of 10	Prefix	Symbol	Power of 10	Prefix	Symbol
24	yotta	Y	-1	deci	d
21	zetta	Z	-2	centi	c
18	exa	E	-3	milli	m
15	peta	P	-6	micro	m
12	tera	T	-9	nano	n
9	giga	G	-12	pico	p
6	mega	M	-15	femto	f
3	kilo	k	-18	atto	a
2	hecto	h	-21	zepto	z
1	deka	da	-24	yocto	y

Many of the prefixes come from Greek and Latin words, often via French. "Zepto" is derived from the Latin septem, meaning 7, because this is the seventh prefix in the system of metric prefixes. The s was replaced by z to avoid confusion with the abbreviation for the second. The prefix "zetta" was coined to parallel "zepto." Similarly, "yocto" is derived from the Latin octo, meaning 8, and "yotta" parallels that term.

You can find a guide to units of measurements and their history at A Dictionary of Units of Measurement.

You might be interested to know that an attoparsec is a distance of about one inch (3.1 centimeters). The distance for Earth of the most remote object yet observed in the universe is about 125 yottameters (13.2 billion light-years). The diameter of the largest known galaxy is about 53 zettameters. At the other end, an atomic mass unit equals 1.66 yoctograms.

The Convert Auto page (a student project) offers a handy tool for converting from one unit of measurement to another in a wide range of fields.

Now you can really start talking ultra large and extra small.

Originally posted July 27, 1998

References:

Hilbert, M., and P. Lopez. The world’s technological capacity to store, communicate, and compute information. Science Express (Feb. 10, 2011).

Morrison, P., and P. Morrison. 1998. Wonders: The sum of human knowledge? Scientific American 279(July):115.

Peterson, I. 1993. Measuring superconductor magnetic noise. Science News 143(Jan. 16):37.