The Social Applications of Highly Composite Numbers:
Efficiency, Self-Organization, Emergence and Economics
by Bill Lauritzen
There is an important, yet simple, class of numbers, perhaps equal or greater in importance to prime numbers, that has remained relatively hidden over the centuries, but has a great deal of social usefulness. Mathematicians call these numbers “highly composite numbers.” (I use to call them "versatile numbers," but I will mostly use the traditional name here.) Some famous modern mathematicians have investigated these numbers, and possibly one Greek philosopher.
These are the numbers that were used by many of the ancients to mentally pattern the universe around them: the circle of the horizon, time, weight, length, classes of items, and even numbers themselves. These are the numbers that answered these fundamental questions: How many? How much?
How should one divide up space, time, matter, and energy for measuring? How should one pack numbers together into groups? (In other words, what "base" should be used, or, how should the society divide up "infinity.")
Homo sapiens apparently has a desire to split up the universe around him, in order to grasp it better, so he uses the mental tool of numbers. This mental tool overlays a pattern on the universe, but also act as filter-glasses for seeing the universe. We see the world through the mental filter-glasses of 10, which is not a highly composite number.
This paper will emphasize the more practical aspects of highly composite numbers, rather than the purely mathematical aspects of these numbers. (See the references for articles on that.) Even so, a full book could be written about these numbers, and I will here try only to summarize, I fear in an entirely inadequate manner.
My hypothesis is that knowledge of this class of numbers has, in the past, lubricated social and economic action between people. As the world becomes increasingly populated, creating greater and greater social stress, greater knowledge and use of this class of numbers by the general public might further lubricate economic interaction among people. Unfortunately, with the rise of metric measurement, it appears that just the opposite is occurring.
My fundamental thesis can be stated on one sentence: Liberal use of highly composite numbers (2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, ...) would lubricate a human’s interaction with surrounding humans and the surrounding environment, especially as population on Earth increases.
2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, ... (In mathematical language, n is highly composite if f(n) > f(x) where f(n) and f(x) are the number of factors of n and x, for all x < n. )
We could say prime numbers have a minimum number of factors while highly composite numbers have a relative maximum of factors. As one mathematician, Hardy, who we will meet later, said, “they are as unlike a prime as a number can be.” (Kanigal, p. 232)
If we compare the number 12 with the number 23, along the lines of addition, we see that the number of ways to split the numbers into two integer parts is generally determined by the size of the number. In other words, 23 can be written as 1+ 22, 2 + 21, 3 + 20, 4 + 19, 5 + 18, 6 + 17, 7+ 16, 8 + 15, 9 + 14, 10 + 13, and 11 + 12. Whereas 12 can be written only as 1 + 11, 2 + 10, 3 + 9, 4 + 8, 5 + 7, and 6 + 6.
However, if we were to express 12 and 23 as the products of two numbers rather than the sums of two numbers, an entirely different story emerges. Twenty-three can be written as 1 x 23 only. Twelve can be written as 1 x 12, 2 x 6, and 3 x 4. The smaller number can be split in more ways. We say 23 has two "factors," while 12 has six. Twelve is more versatile than 23.
Here's a sample factor table: In the first column we have the number, in the second column we have a list of all the factors of the number, and in the third column we have the number of factors of the number.
|number||divisors||number of divisors|
Notice that this third column is rather erratic. It's this "erraticness" that allows us to pick out those numbers that have more or an equal number of factors compared to the numbers around them.
EXAMPLE: 12 is highly composite because with 6 factors (1, 2, 3, 4, 6, 12), it has more factors than all the smaller numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11.
A highly composite number is a number that has a greater number of factors than any smaller number. Whenever the number of factors from the list above jumps, we designate a highly composite number. Here are the first few highly composites with the number of factors is in parentheses: 2 (2), 4 (3), 6 (4), 12 (6), 24 (8), 36 (9), 48 (10), 60 (12), 120 (16), 180 (18), 240 (20), 360 (24), 720 (30), 840 (32), 1260 (36), 1680 (40), 2520 (48), 5040 (60), ...
There is no way to predict the next one except by trial. In other words, it's fascinating to try to look for patterns in these, but, there are none. For a while I thought that there was a prime next to every highly composite except for 120. [This holds true up to 25,200 (90).]
So highly composite numbers, like primes, can not be predicted by any formula. Another characteristic that they share with primes is that they become less frequent as they get larger.
Versatile numbers are a sort of potential numerical nexus point. A point where many numbers can meet.
Steven Ratering of Central College wrote, in 1991, a paper about “highly composite numbers” which he called by the name “round numbers.” He felt, and I believe rightly so, that these numbers were “rounder” than 10, 20, 30 ... 100, etc.
What use are highly composite numbers?
It’s hard to say exactly what percent of human interaction is involves numbers. Trade (economics and business), construction of shelter, and sports certainly take up a large proportion of human time.
EXAMPLE: A school teacher has 23 students in her class, a non-highly composite number. If she wants to divide the students into groups, she could, but the groups would never have the same number in each. (She can't take a fraction of a student.) If she had 24 students, however, a highly composite number, then she could divide them into groups of 2, groups of 3, groups of 4, groups of 6, groups of 8, or groups of 12, all with exactly the same number in each group.
EXAMPLE: Merchant A imports 360 items, a highly composite number. Merchant B imports 375 items, a non-highly composite number. The 360 items can be divided into 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, or 360 even groups. The 375 items can only be split into 1, 3, 5, 15, 25, 75, 125, and 375 even groups, unless fractions of items are used.
EXAMPLE: Some children have some apples to divide equally between them. If they have 12 apples, a highly composite number, the apples could be shared equally with 2, 3, 4, or 6 children. No bloody noses. If they have 13 apples, a non-highly composite number, the apples can not be shared equally. Unless they know how to make fractions of items quickly and easily, bloody noses are possible. (I have done some preliminary experiments along these lines, which suggest that this phenomenon is worth investigating.)
EXAMPLE:A real estate broker can divide up land into lots of 25 acres. This means one could split it into 5 smaller lots of 5 acres each. To divide it up into a highly composite 24 acres means one would have many more options: 2 by 12 acre lots, 3 by 8 acre lots, 4 by 6 acre lots.
EXAMPLE: The state legislature wants to reduce class size. What number do they pick? The highly composite numbers are 36, 24 and 12. This numbers should be given most consideration as they are the easiest to divide up.
There are an infinite number of such examples. How many items should one export? How many items should one manufacture? How many items should one pack together? How many people, states, districts, counties, etc., should there be?
These are the many common decisions that one faces on the job, in which one has to pick how many. Thus, unless one is working in some strictly mathematical job, one is much more likely to have a need for versatile numbers than prime numbers.
I have had many jobs besides being a school teacher. Over twenty. On none of these jobs was knowledge of prime numbers necessary. However, on most of the jobs I was required to make a decision about how many. In other words, a good case could be made that it is more important to teach highly composite numbers in school than it is to teach prime numbers.
To the average person fractions of items are not that easy to deal
with. (There are two types of fractions: fractions of an item, such as
1/3 of the apple, and fractions of a total, such as 1/3 of all the
apples.) They take time and effort, especially when you are dealing with
decimal fractions that are repeating such as 0.33333.... As we can see
from the above examples, sometimes fractions of items are unnecessary if
one chooses to work with a highly composite number. Fractions cause
unnecessary stress. And unnecessary stress can have detrimental
psychological and physiological effects.
This is the proposition I am making: that the use of particular
numbers could cause less stress and could lubricate economic and social
interactions. It could be tested by psychologists. Perhaps all of us can
recall a time in their life when there was an upset because someone got
more than we did.
Will there be more antisocial behavior among this group of children if they have to divide up 10 pieces of candy or 12 pieces of candy? Put some children in a room (a small group of 2 to 6 children as is commonly seen) and give them either 12 or 10 pieces of candy and let them figure out how to share the candy (similar to the above example with apples). With 5 children there may be more antisocial behavior with the 12 candies. However, with 2, 3, 4, and 6 children I predict that with the 10 pieces of candy there will be more antisocial (aggressive) behavior.
Another proposition is that the mere awareness of a certain class of numbers by a civilization could increase the intelligence of that civilization. Again, this could be tested by psychologists. Make two groups of school children who are matched in mathematical ability (Group A and Group B). Teach Group A mostly about prime numbers, in the traditional manner, while Group B is taught about both prime and highly composite numbers. Then administer standard math tests to both groups. I predict the group taught highly composite numbers would test higher.
It may be a small and subtle advantage that the person or culture dealing with highly composite numbers has, but in life, even a small advantage, over time, can lead to the extinction of competitors. Remember that chimps have something like 99.9% of their active DNA in common with humans. Look at what difference that small advantage can make. I believe this same mechanism operates in this situation. In other words, a business with a small advantage (such as the liberal use of highly composite numbers) will, other things being equal, have a better chance of survival.
The fact is that it is easier to share or distribute evenly using highly composite numbers than any other kind of numbers. Why is even sharing or distribution important? Remember that nature does two crucial tasks. One is to bring things together. This could be through gravity or though human packing of goods. The second is to spread things out. This could be through energy radiation or through human distribution. So packing, and its opposite, distribution, are vital to understand. For example, if there is a shortage of something (such as food) on one side of the globe and a surplus of the same thing on the other, it is to humanity's advantage to be able to pack, transport, and then distribute this food efficiently, easily, and evenly.
As a computer programmer, John Boyer, pointed out to me, primes are used in data encryption, in other words, to keep data secret, or prevent its distribution. Sort of the opposite of what I am recommending for highly composite numbers.
Don't get the idea that I favor a society in which everyone gets the same reward regardless of the amount of work they do. Sometimes it is important to be able to share unevenly. (In that case a larger number can be shared in more ways.) However, the situations in which even sharing is desirable are more widespread.
So I believe that school children should be able to define and list highly composite numbers, just like they do prime numbers. This will give them insight into the character of numbers. They should also be taught to use highly composite numbers in real situations as mentioned at the beginning of this article. Merchants, politicians, businessmen, legislators, in fact, all citizens could benefit from knowing these numbers. In other words, we should work to make these numbers part of the standard curriculum for all schools.
Babylon and Highly Composite Numbers
Three highly composite numbers (12, 60, and 360) were ones that the Babylonians chose near the dawn of civilization to divide up the heavens (360 degrees), the circle (360 degrees), time (12 hours--the Babylonian day had 12 hours not 24), more time (60 minutes and 60 seconds), and their number system (base 60).
This base 60 number system has always been a mystery and we find Oystein Ore (Number Theory and its History) writing, "It is difficult to explain the reasons for such a large unit group."
Why did the Babylonians picked these groupings? One hypothesis is that they got them from astronomy. However, note that 365.25 (days per year) and 12.4 (lunar months per year) are the only astronomical numbers close to highly composite numbers. A strict astronomical hypothesis, I think, is wrong. I suggest that the Babylonians chose 12, 60, and 360 partly because of the closeness of 12.4 and 365.25 and partly because these numbers have relatively large numbers of factors. In other words, it's possible that the Babylonians were aware of the class of numbers I call highly composite numbers.
It may be a very fortuitous astronomical circumstance that we have 12.4 months and 365.25 days per year. The closeness in size of 12.4 and 365.25 to highly composite numbers may have led to early humans being made more aware of this class of numbers.
It is amazing to me how many people believe that our time system was handed down by God and can not be changed.
When I say to people, “We don’t have to have 24 hours in a day,” they say, “Yes we do, because that’s how many hours there are in a day.” In other words, they believe that these numbers are set by nature. Actually, the only one that is set by nature is the 365.25 days in a year (approximate), as that’s how long it takes the Earth to go around the sun. (Although 12.4 lunar months in the year is close to 12 months per year.) The other numbers are somewhat arbitrary. In other words, instead of 24 hours in a day, we could have 15 kunas in a day. And 50 kinas in a kuna, and 100 kinitas in a kina, or whatever your imagination could construct.
In am not suggesting that we restructure our time system. The 12, 24, 60, and 60 are all highly composite numbers and were perhaps chosen in part for that very reason. However, given the present metric fashion, one would expect that legislators would next try to make 100 minutes in an hour, etc. So perhaps its good that people think these time numbers are set by nature.
The Old Paradigm
Traditional mathematics has divided numbers into "abundant numbers, perfect numbers, and deficient numbers." They are defined as follows:
1) abundant number: the sum of the factors of a number, except for itself, is greater than itself. The first few are: 12 (6), 18 (6), 20 (6), 24 (8), 28 (6), 30 (8), 36 (9), 40 (8), 42 (8), 48 (10), 54 (8), 56 (8), 60 (12), 66 (8), 70 (8), 72 (12), 78 (8), 80 (10), 84 (12), 88 (8), 90 (12), ... These abundant numbers are somewhat similar to highly composites but are much less exclusive.
2) perfect number: the sum of the factors of a number, except for itself, equals itself. The first few are: 6, 28, 496, 8128, 33 550 336, 8 589 869 056, 137 438 691 328, ...
3) deficient number: the sum of the factors of a number, except for itself, is less than itself: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, .... This includes all the numbers not listed in the first two definitions.
(Many mathematicians are aware of a class of numbers called
“superabundant numbers” which I do not go into here. Highly composite numbers
do not correspond to superabundant numbers except at the very beginning
of the series.)
When and why did the "abundant, perfect, deficient" paradigm begin? Euclid, around 300 BC defined a perfect number as "that which is equal to its own parts."
Nicomachus, around 100 AD, stated that all odd numbers were
deficient. (He was wrong; 945 is abundant.) He discussed "even abundant"
and "even deficient" numbers. He compared "even abundant numbers" to an
animal with "too many parts or limbs, with ten tongues, as the poet
says, and ten mouths, or with nine lips, or three rows of teeth ...". An
"even deficient number" was said to be as though "one should be one
handed, or have fewer than five fingers on one hand, or lack a tongue
...". Perfect numbers, he said, are akin to "wealth, moderation,
propriety, beauty, and the like ...".
All in all, not a very scientific analysis.
In more recent times, L.E. Dickson, in 1952, (in the classic History of the Theory of Numbers, 3 vols.), gives an extensive history of number theory, with a complete documentation of names and dates--except for abundant, perfect, and deficient numbers. The book was written as if these three categories had always existed, or had been handed down from some divine entity. But they must have started somewhere, and for some reason.
Some Mental Patterns
End Part I of III
Highly Composite Numbers
Egypt and the Old Paradigm
Richard Friedberg, in 1968, (An Adventurer's Guide to Number Theory) implied that Pythagorus, around 600 BC, knew the three classes of abundant, perfect, and deficient numbers, and suggested that they developed because of the way the Egyptians wrote fractions. They never wrote 11/12. Instead they would write 1/2 + 1/3 + 1/12, never putting anything but a "1" in the numerator.
Also, they never used the same denominator more than once. As a
result, all the perfect numbers can be split up "perfectly." Six can be
split into 1/2 + 1/3 + 1/6 or 6/6. Twelve can be split up into 1/2 +
1/3 + 1/4 + 1/6 + 1/12 or 16/12. Twelve is "abundant." However, 10 can
be split up only into 1/2 + 1/5 + 1/10 or 8/10. It's a "deficient"
If Friedberg was correct, in our own number system, using our fractions “perfect” numbers are not necessary. In other words, the names “abundant,” “perfect,” and “deficient”, and the paradigm they represent, may be an anachronism.
Highly Composite Numbers on the Internet
When I posted information on the internet regarding highly composite numbers, I received more e-mail than I could keep up with. It came from the United States, France, Netherlands, Germany, and Russia, from mostly people who are much better mathematicians that I. It contained conjectures, proofs, computer generated lists of highly composite numbers, and just pure speculations about numbers.
One advanced school in Russia, the Math Center of the Palace of Youth Creativity, had a “Versatile Number Day,” at the instigation of mathematics teacher Roman Breslav (firstname.lastname@example.org). They proved some things concerning highly composite numbers and made several conjectures. I think this activity was remarkable, and this school is undoubtedly way ahead of most.
Plato and Highly Composite Numbers
One day I received e-mail from a Ph.D. mathematician in Switzerland, Meyer Rainer, who pointed out a passage in Plato’s work that he claims suggests strongly that Plato knew of highly composite numbers. I am not a Greek scholar, so it is difficult for me to judge the validity of Dr. Rainer’s claim, but it sounds plausible. The passage is on pages 746-747 of Laws V, where he says,
There is no difficulty in perceiving that the twelve parts admit of the greatest number of divisions of that which they include, or in seeing the other numbers which are consequent upon them ... the divisions and variations of numbers have a use in respect of all the variations of which they are susceptible, both in themselves and as measures of height and depth, and in all sounds, and in motions, as well those which proceed in a straight direction, upwards or downwards, as in those which go round and round. The legislator is to consider all these things and to bid the citizens not to lose sight of numerical order; for no single instrument of youthful education has such mightily power, both as regards domestic economy and politics, and in the arts, as the study of arithmetic. ...if only the legislator ... can banish meanness and covetousness from the souls of men ... [my emphasis]
I was amazed to find Plato apparently discussing some of the same issues concerning numbers as I had been interested in. This appears to me to be the first attempt since ancient Babylon to utilize the versatility of these numbers in a social setting. Here are some additional relevant remarks from Laws V, p. 737-738, that Dr. Rainer drew my attention to:
freedom from avarice and a sense of justice--upon this rock our city shall be built; for there ought to be no disputes among citizens about property ... that [the people] should create themselves enmities by their mode of distributing lands and houses, would be superhuman folly and wickedness. How then can we rightly order the distribution of the land? In the first place the number of the citizens has to be determined, and also the number and size of the divisions into which they will have to be formed; and the land and the houses will then have to be apportioned by us as fairly as we can ...The number of our citizens shall be 5040--this will be a convenient number ... Every legislator ought to know so much arithmetic as to be able to tell what number is most likely to be useful to all cities; and we are going top take that number which contains the greatest and most regular and unbroken series of divisions. The whole of number has every possible division, and the number 5040 can be divided by exactly fifty-nine divisors [sixty including itself], and ten of these proceed without interval from one to ten; this will furnish numbers for war and peace, and for all contracts and dealing, including taxes and divisions of the land. These properties of numbers should be ascertained leisure by those who are bound by law to know them; for they are true, and should be proclaimed at the foundation of the city, with a view to use. [my comment]
Plato considered the highly composite number 5040, according to Dr. Rainer, as "an ideal number of citizens in an ideal community, where everyone lives in peace, freedom, and friendship, and all measurements, weightings, and partitions are done in the proper way."
Ramanujan and Highly Composite Numbers
Coming forward two thousand and three hundred years, we find an article published in 1915 (Proceedings of the London Mathematical Society, Vol. 14) in which the noted Indian mathematician Srinivasa Ramanujan analyzes what he calls "Highly Composite Numbers." This paper is now considered a classic.
Ramanujan was a fascinating character and much has been said about him in articles, books, and documentaries. Nova (WGBH, Boston) had a documentary about him called “The Man Who Loved Numbers.” A book was written about him: The Man Who Knew Infinity, by Robert Kanigel. He had a brief but brilliant life.
Ramanujan’s only exposure to modern European mathematics (of his time) was one book on mathematics. He single-handedly re-derived some 1915 mathematics, and a good deal more, by himself. Scientists and mathematicians today are still finding new meaning in his work.
Ramanujan was always looking for new ways to do things. He may not have known of the traditional mathematical paradigm (of abundant, perfect, and deficient numbers). As he said in his famous letter to G .W. Hardy (the brilliant British mathematician who brought Ramanujan to England), in 1913, "I have not trodden through the conventional regular course which is followed in a University course, but am striking out a new path for myself."
Here's his definition of a "highly composite number": "I define a highly composite
number as a number whose number of divisors exceed that of all its predecessors." This is the same as a highly composite number. (In mathematical language: the number n is called highly composite if d(m) < d(n) for all m < n where d(n) is the number of divisors of n. “Divisors” here is synonymous with “factors.” )
Let me give you some idea of the magnitude of his mathematical genius. Without the use of a computer, Ramanujan had calculated all the highly composites up to 6 746 328 388 800 (10 080 factors). He only missed one.
With regard to predicting highly composite numbers he came to a similar conclusion to mine: "I do not know of any method for determining consecutive highly composite numbers except by trial."
It's true that every composite number can be expressed as the product of primes. In one sense, primes are the raw building material of the other numbers. But what good is the building material without the building? What good are the chemical elements without the compounds? What good are the organs of the body without the body?
Do we call our great cathedrals, temples, skyscrapers, geodesic domes, and other works of architecture merely "composites"? In other words, do we primarily (no pun intended) study our shelter materials and secondarily our shelters? Or, should we be most concerned with our shelters, and as a result, be interested in what they are of made of?
By the end of this essay I hope to have convinced you more fully that highly composites are as important as primes. "Prime" usually implies some excellence or value. The word puts undue emphasis on these numbers.
The term "highly composite" might be descriptive to someone
trained in mathematics, however, I believe the term “versatile” is
simpler yet still descriptive, and should be used in order to
communicate to the largest number of people the character and usefulness
of these numbers. What would you think if I called prime numbers
“minimally composite numbers?” How many people
know what a “Schwarzschild singularity” is? Not many. Yet, how many
people know what a “black hole” is? Almost everybody. That’s because
physicist John Wheeler would often meditate for months in order to find
just the right name for something. Nowadays, the name “black hole” is
In the mathematical world, we find a another famous mathematician, Paul Erdös (pronounced erdish), involved with highly composite numbers
Erdös was an interesting character that used to travel around the world with just a small suitcase, living with other mathematicians, and doing math with them. He died in 1997. A book was written about him: The Man Who Loved Only Numbers.
Erdös wrote about “highly composite numbers” in 1944. Later he wrote about them with L. Alaoglu and with French mathematician Jean-Louis Nicolas. Anyone who wants to delve into the purely mathematical nature of highly composite numbers should consult works by Erdös and these people. (See references.)
Artificial Intelligence Discovers Highly Composite Numbers
Doug Lenat, one of the foremost researchers in Artificial Intelligence, wrote a program called AM (Automated Mathematician) when he was at the Stanford AI Lab in 1976. AM "discovered" many concepts of standard number theory. It was programmed such that if it discovered something interesting, it should also investigate its inverse. Thus after it "discovered" prime numbers it also looked at numbers having a maximum of primes. At first Lenat thought AM had discovered something completely original, but he later read about Ramanujan's work.
There are a limited number of Platonic solids: the tetrahedron (which I call the four-corner or the four-nook), the octahedron (six-nook), the cube (eight-nook), the icosahedron (twelve-nook), and the dodecahedron (twenty-nook). Notice that many of the number of corners, edges, and faces of these figures are highly composite numbers or numbers with relatively large numbers of factors.
Five Platonic Solids
It was partly due to my study of these figures, which was inspired by Buckminster Fuller, that I discovered highly composite numbers on my own.
Buckminster Fuller introduces a class of numbers somewhat related to highly composite numbers which he called Scheherazade numbers. Although he never formally defines these, we can glean the fact that they are equal to the product of primes. (He called these Scheherazade numbers as the prime numbers 7 x 11 x 13 equal 1001 and Scheherazade was a character in One Thousand and One Nights.) Mathematicians call these numbers "prime-factorial" or "primordial" numbers. As examples, 1 x 2 x 3 x 5 gives the primordial number 30, and 1 x 2 x 3 x 5 x 7 x 11 x 13 gives the primordial number 30030.
However, the lower highly composite number 24 has as many
30. And the primordial number 30030 has 64 factors, while the closest
highly composite number, 27720, a lower number, has 96 factors, or 32
factors. Although primordial numbers are more encompassing with regard
to primes, highly composite numbers are more encompassing with regard to