Earth360 Home > Articles> Axioms of Design 

Versatile Numbers:
SelfOrganization, 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,” but here I call them "versatile numbers." 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 filterglasses for seeing the universe. We see the world through the mental filterglasses of 10, which is not a versatile number.
This paper will emphasize the more practical aspects of versatile 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 versatile 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.
Definition
2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, ...
(In mathematical language, n is versatile 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 versatile 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. Twentythree 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.
FACTOR TABLE
number  divisors  number of divisors 

1  1  1 
2  1,2  2 
3  1,3  2 
4  1,2,4  3 
5  1,5  2 
6  1,2,3,6  4 
7  1,7  2 
8  1,2,4,8  4 
9  1,3,9  3 
10  1,2,5,10  4 
11  1,11  2 
12  1,2,3,4,6,12  6 
13  1,13  2 
14  1,2,7,14  4 
15  1,3,5,15  4 
16  1,2,4,8,16  5 
17  1,17  2 
18  1,2,3,6,9,18  6 
19  1,19  2 
20  1,2,4,5,10,20  6 
21  1,3,7,21  4 
22  1,2,11,22  4 
23  1,23  2 
24  1,2,3,4,6,8,12,24  8 
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 versatile 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 versatile 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 versatile number. Here are the first few versatiles 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 versatile except for 120. [This holds true up to 25,200 (90).]
So versatile 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 versatile 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 nonversatile 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 versatile 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 versatile number. Merchant B imports 375 items, a nonversatile 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 versatile number, the apples could be shared equally with 2, 3, 4, or 6 children. No bloody noses. If they have 13 apples, a nonversatile 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 versatile 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 versatile 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 versatile 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 versatile 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.
Child A >  Child A >  
Child B >             Child B >              
Child C >  Child C > 
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 versatile numbers. Then administer standard math
tests to both groups. I predict the group taught versatile numbers
would test higher.
It may be a small and subtle advantage that the person or culture
dealing with versatile 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 versatile 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
versatile 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 versatile 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 versatile numbers, just like they do prime numbers. This
will give them insight into the character of numbers. They should
also be taught to use versatile 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.
Three versatile 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 hoursthe 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 versatile 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 versatile
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 versatile 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.
OUR TIME NUMBERS: 
NATURE’S NUMBERS: 
365.25 days/year 
(mandated by nature) 365.25 days per year 
12 months/year 
(suggested by nature & versatile) 12.4 lunar months per year 
7 days/week 
approximately a halfmoon (7.36 days) 
24 hours/day 
(compatible with 12 and 360 & versatile) 
60 minutes/hour 
(compatible with 12 and 360 & versatile) 
60 seconds/minute 
(compatible with 12 and 360 & versatile) 
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 versatile 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.
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 versatiles 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. Versatile numbers do not correspond to superabundant numbers except at the very beginning of the series.)
Abundant number: 12 
because 1 + 2 + 3 + 4 + 6 = 16 which is greater than 12. 
Perfect number: 6 
because 1 + 2 + 3 = 6 which equals 6. 
Deficient Number: 10 
because 1 + 2 + 5 = 8 which is less than 10. 
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 datesexcept
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.
infinity  ... 
mental pattern imposed: group by  (suggested by  fingers) 
infinitely increasing time: 
mental patterns imposed (partly
suggested by our solar sytem): year: mandated by 365.25. months: suggested by 12.4 lunar cycles. others: (hrs, minutes, seconds) made to be compatable with year, month, also versatile. 
infinitely increasing weight: 
mental pattern imposed: pound 
infinitely increasing measure: length 
mental pattern imposed: 12 inch3 foot1 yard 
infinitely divisable circle of the horizon: 
mental pattern imposed: 360 divisions (degrees) 
infinitely increasing temperature: 
mental pattern imposed: 100 divisions between freezing and boiling point of water (C). (points suggested by nature) 
End Part I of III
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" number.
Abundant Number: 12 
because 1/2 + 1/3 + 1/6 + 1/12 = 16/12 
Perfect Number: 6 
because 1/2 + 1/3 + 1/6 = 6/6 
Deficient Number: 10 
because 1/2 + 1/5 + 1/10 = 8/10 
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.
When I posted information on
the internet regarding versatile numbers, I received more email
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 versatile 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 (roma@rbb.stud.pu.ru).
They proved some things concerning versatile numbers and made
several conjectures. I think this activity was remarkable, and
this school is undoubtedly way ahead of most.
One day I received email 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 versatile 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 746747 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. 737738,
that Dr. Rainer drew my attention to:
freedom from avarice and a sense of justiceupon 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 5040this 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 fiftynine 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 versatile 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."
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 singlehandedly
rederived 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 versatile 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 versatiles up to 6 746 328 388 800 (10 080 factors). He
only missed one.
With regard to predicting versatile 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 versatiles 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?”
I’m sure some mathematicians resent the fact that I am trying
to change the “standard” nomenclature from “highly
composite” to “versatile.” Perhaps they need a
lesson from physics: 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 immediately recognizable.
So I am primarily interested in defining numbers that the public,
legislators, and an average 12 year old student can easily remember
and use.
In the mathematical world,
we find a another famous mathematician, Paul Erdös (pronounced
erdish), involved with versatile 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 JeanLouis Nicolas. Anyone who wants to delve into
the purely mathematical nature of versatile numbers should consult
works by Erdös and these people. (See references.)
Artificial Intelligence Discovers Versatile 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.
Platonic Solids
There are a limited number of Platonic solids: the tetrahedron (which I call the fourcorner or the fournook), the octahedron (sixnook), the cube (eightnook), the icosahedron (twelvenook), and the dodecahedron (twentynook). Notice that many of the number of corners, edges, and faces of these figures are versatile numbers or numbers with relatively large numbers of factors.
name 
corners 
edges 
faces 
tetrahedron (4nook)  4  6  4 
octahedron (6nook)  6  12  8 
cube (8nook)  8  12  6 
icosahedron (12nook)  12  30  20 
docedahedron (20nook)  20  30  12 
It was partly due to my study of these figures, which was inspired by Buckminster Fuller, that I discovered versatile numbers on my own.
Buckminster Fuller introduces
a class of numbers somewhat related to versatile 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 "primefactorial"
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 versatile number 24 has as many factors as
30. And the primordial number 30030 has 64 factors, while the
closest versatile number, 27720, a lower number, has 96 factors,
or 32 more factors. Although primordial numbers are more encompassing
with regard to primes, versatile numbers are more encompassing
with regard to all numbers.
Primordial Numbers:
2, 6, 30, 210, 2310, 30 030, 510 510, 9 699 690, ... 
Versatile Numbers: 2,
4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, ... 
Although Ramanujan brought
these numbers to the attention of the mathematical community,
has these numbers found their way into the mainstream of society?
Have the public and legislators used these numbers to bring about
ease of computation, packing and distribution? The answer is "no,"
(which is why I am writing this paper).
Hardy, (see Collected Papers of G. W. Hardy, Vol. VII), called
Ramanujan's paper "... the largest and perhaps the most important
connected piece of work which he has done since his arrival in
England." Although Ramanujan died in 1920, at a young age,
Hardy went on to live many more years, until 1947, and continued
to study Ramanujan’s work. I wondered why Hardy did not try
to apply this work to society. Hardy himself answered this question:
"If asked to explain how, and why, the solution of the problems
which occupy the best energies of my life is of importance to
the general life of the community, I must decline the unequal
contest ... A pure mathematician must leave to happier colleagues
the great task of alleviating the sufferings of humanity."
(quoted in The Man Who Knew Infinity, p. 347)
Versatile numbers have been studied rather extensively by mathematicians.
But perhaps they have been buried in with mountains of other mathematical
facts, which may or may not have any practical value.
I think it is time that economists, psychologists, sociologists,
legislators, and others become aware of these numbers.
The ancient merchants possibly intuitively knew to use a versatile 12, or dozen, in their trade. There are many other examples of the ancients using a versatile dozen or a close relative.
SOME DOZENAL EXAMPLES 
twelve inches to a foot 
twelvepack 
sixpack 
twelve parts to the chromatic scale 
twentyfour to a case 
two times twelve hours in a day 
a dozen twelve to a gross 
twelve hours in a Babylonian day 
twelve ounces in a troy pound 
twelve lines to an inch 
twelve pence to a shilling 
twelve months to a year 
two times twelve letters in the Greek alphabet 
twelve, twentyfour, and thirtysix to a roll of film 
A base twelve numbering system
was proposed in 1586 by Simon Stevin, and again in 1760 by Georges
Louis Leclerc.
Then, in October of 1934, an author by the name of F. Emerson
Andrews, wrote an article for The Atlantic Monthly, “An Excursion
in Numbers,” which eventually led to the formation, in 1944,
of the Duodecimal Society (later called the Dozenal Society).
The Society rightly points out that a dozenal base is better than
a ten base. It has investigated the mathematics of a dozenal system.
It also publishes papers concerned with dozenal systems. I was
the annual guest speaker for the Dozenal Society in 1997 and gave
a lecture on “Numbers of the Future?,” a base twelve
number system.
Isaac Asimov was a member of the Dozenal Society and has several
paragraphs in Realm of Numbers concerning the advantages of a
base twelve number system, but seems convinced that our ten fingers
are too big an obstacle to overcome. This point might be refutable
by simply counting with the thumb on the three bony segments of
each four fingers, which gives a versatile twelve.
Despite a few dissenters, the world has continued to slip toward
base ten and the socalled scientific base ten metric system.
For example, in 1971, the British switched from halfpennies,
pennies, threepence, sixpence, shillings, halfcrowns, pounds,
and guineas (a 1/2, 1, 3, 6, 12, 30, 240, 252, system which uses numbers with
lots of factors or versatile numbers), to a decimal (1, 5, 10,
20, 50, 100, 1000, etc.) mostly nonversatile, monetary system.
The more versatile numbers were in use when London was the foremost
financial center of the world. The English system of measurement
with 12 inches to the foot was in use when the U.S. put a man
on the moon. Now the U.S. is trying (with much resistance) to
go to a nonversatile metric system.
It's a noble goal to align all your measuring systems with your
number system, and those who have tried to do so should be thanked
for their efforts. However, instead of changing our versatile
measuring system to match our numbers, perhaps we should have
changed our numbers to match our more versatile measuring system.
Perhaps we really just got off on the wrong foot (maybe I should
say hand) when we started counting by tens, using our eight fingers
and our two thumbs.
Why haven't versatile numbers,
in a sense the "shelters," the "great cathedrals,"
the “marvelous organisms” of all our numbers, been more
intensively taught to the general public? I believe we've been
completely surrounded by versatile numbers for so long (360 degrees
in a circle, 360 degrees in the heavens, 12 months, 60 seconds,
60 minutes, 24 hours, 12 in a dozen, 12 inches in a foot, 6packs,
12packs, 122436 pictures in a roll of film, 12 notes, 12 pence,
etc.) that we have forgotten about them. It’s like the story
of two fish having an argument about whether water exists.
And even if someone does discover versatile numbers, they are
an embarrassment to a society that uses "ten," not a
versatile number, as the core of its number system. One can group
numbers by  (10, as we do), by  (12, as
I suggest), or by  (20, as the Mayans
did), or by 
(60, as the Babylonians did), or whatever you want.
HinduArabic (10)        
Dozenal (12)        
Mayan (20)     
Babylonian (60) 
 
A base 10 system (or a base
12 system) overlays a mental pattern on numbers.
In fact, all patterns in numbers (other than 1 + 1 + 1 ...) are
the result of the versatility or factorability of numbers. For
example, when one looks for patterns in number, prime numbers
have no pattern in them, while versatile numbers have the most
patterns. In other words,  (12) can be composed like
this:   or     or    or 
    . Whereas with  (13), you can’t
make any patterns (except that basic pattern underlying all number
 +  +  +  +  ...).
 no patterns 
One interesting possibility
would be to have a counting system that kept changing to the next
versatile number base. Our time system, which shifts from a 60
second and 60 minute base to a 24 hour base is somewhat like this.
I think it might be possible.
The mental number pattern that Homo sapiens imposes on the world
results in everyone looking at the world through a pair of glasses
that could be called, “baseten glasses.” These glasses
will tend to filter out certain things, and let in other things.
Someone with these glasses would see 
and say, “I see ten and ten and four, or 24.” Someone
with more versatile basetwelvefilterglasses would say, “I
see twelve and twelve, or two dozen.”
 Twelve as a pure amount or number without base10 filter glasses. 
(  +  ) + (  +  ) + (  +  ) Breaking twelve down into its inherent no patterns (primes) , , and . 
            Breaking this down further into the most fundamental material of twelve,  +  +  ..., we sort of arrive back where we started. 
Thus, when one has to make
a common decision about how many, as in the example as I mentioned
earlier, in which the state legislature wants to reduce class
size, perhaps they will have choices of 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30. The basetentoolfilterglasses will
tend to make them see, and thus chose, 20, 25, or 30, when the
most versatile and easy to divide number in this range is 24.
Versatile basetwelvefilterglasses would have immediately shown
the versatile number 24. Practical filterglasses would have shown
18, 20, 24, and 30.
Thus, in the long term, perhaps our numbering system should have
a versatile number at its core rather than a nonversatile number.
In my booklet, Nature's Numbers, and in my video, “Numbers
of the Future,” I discuss this, and also propose what I believe
is a more efficient, easily learned numbering system (not using
09 but entirely different symbols of my own invention) derived
from versatile numbers. I have personally taught this system to
over 6000 students.
In my video of 1995, I relate the versatile number 12 to easily remembered geometrical modelsmodels formed due to what I call “the gravitational accumulation of spheres” (closest packing of spheres). I found in my research that the gravitational accumulation of spheres results in only five regular shapes consisting of: two balls, three balls, four balls, six balls, and twelve balls. No matter how many balls you add to the total, you can only get these six regular shapes.
Symmetry is just a fancy name for what I call “turnsame.”
If you can turn something and it looks the same as before, that’s
a symmetry of 2. Twelve balls gravitationally packed together,
so that they can not get any closer to each other, has a turnsame
of 60. The symmetries of the two balls, three balls, four balls,
six balls, and twelve balls are 4, 6, 12, 24, and 60all versatile
numbers.
I also found that the symmetry of the system increases up to 12
balls, but no matter how many balls you add to the gravitational
field after that, you can no longer increase the symmetry. Thus,
twelve balls is what I call the “gravitational symmetry limit.”
It’s not surprising to find all these versatile numbers associated
with symmetry when you think about it. Versatile numbers are useful
because they have so many divisors or factors, or potential builtin
patterns. Patterns or repetitions are what make symmetry.
end Part II of III
Building a car that breaks
the world’s land speed record can give insight into automotive
design, and by the same logic, finding the largest versatile can
give insight into numbers.
I am not a computer programmer by trade, but I knew enough to
program my old computer to calculate up to the one hundred and
fifth versatile, V(105). This was just beyond what Ramanujan had
calculated without a computer. He went to V(103). Others have
used computers to calculate versatiles far beyond this.
SOME LARGE VERSATILE
NUMBERS: V (103) = 2^8 * 3^3 * 5^2 * 7^2 * 11 * 13 * 17 * 19 * 23. V (1646) = 2^11 * 3^8 * 5^4 * 7^3 * (11...23)^2 * 29*...241. V (1800) = 2^11 * 3^8 * 5^4 * 7^3 * (11...29)^2 * 31...263. Using the graphing calculator notation, * means “times” and ^ means “to the power of.” The “...” refers to all the prime numbers that have been left out. 
An abbreviated notation is
used for writing these large numbers where “...” means
“all the primes left out.” For example, versatile number
1646, or V(1646) = 2^11 * 3^8 * 5^4 * 7^3 * (11...23)^2 * 29 *...241.
(Using graphing calculator notation here, * means “times”
and ^ means “to the power of.”)
A French paper was brought to my attention, “Methods d’Optimisation
pour un Probleme de Theorie des Nombres,” by Guy Robin, in
which he gives the first versatile number to have more than 10^1000
(that’s a 1 followed by 1000 zeros) factors. This number
is: 2^20 * 3^12 * 5^8 * 7^7 * (11*13*17)^5 * (19*23)^4 * (29...71)^3
* (73...421)^2 * 431...30113. It has 2^3203*3^68*5^2*7*13 factors.
They reported that if we were to write it out it would have 13,198
digits. (However, Guy Robin infomed me recently by email that
it actually has 13,199 digits.)
Two mathematicians I met on the Internet, Noam Shazer and Matt Conroy, brought to
my attention
the method of generating versatiles of huge sizes by using primes.
(See Appendix A.) For example, to calculate V(103) 6,746,328,388,800
one has to only know the primes up to 23. Versatiles of billions
of digits become feasible given simply enough computing time and
computing speed. Although this method can “sling shot”
versatiles from primes, it does not find every versatile.
Jud McCranie holds some records for generating a certain class of prime numbers.
Using this method, he started programming his computer to generate
versatile numbers, and was able to produce a versatile of 7 trillion
digits
There is not a very concise way to write a large versatile, that
I know of, as there is with very large primes. For example a Mersenne
prime with 909,562 digits can be written quite concisely as: 2^3,021,377
 1. (This was the largest known prime, the last time I checked,
but may not be any more. A Mersenne prime is just a prime of the
form “2 to some power minus 1.” So, 2 to the 3rd power,
minus 1, is 7, a prime.)
A computer programmer from Canada, John Boyer,
sent me a proof of the infinitude of versatile numbers. (Appendix D) Since
Euclid proved that primes are infinite, John uses this to show
that versatiles are infinite also.
Primes are essentially inevitable
but unpredictable. In other words, you always know another one
is coming, but you never know exactly where it will appear.
Imagine a Neanderthal who somehow figured out that the first primes
were 2, 3, 5, 7, 11, and 13. Now he is hunting for the next one.
There is no way he can predict when it will arrive unless he tries.
Is it 14? No, that’s divisible by 2. Is it 15? No, that’s
divisible by 3. Is it 16? No, that’s divisible by 2. Is it
17? That’s not divisible by 2, by 3, by 4, or by any other
number. It is 17.
I quote here from Tom M. Apostle, professor emeritus at Cal Tech,
“no simple formula exists for producing all the primes.”
In my thinking, a simple formula involves a pattern, or repetition.
Any repetition implies multiplication. Multiplication implies
a factor. And, if there is a factor involved, then, of course,
we know that the number is not prime. There is no recipe.
This same rule, inevitable but unpredictable, holds true also
for versatile numbers.
They are inevitable, in that they are infinite, and they are unpredictable,
in that we never know when the next one is coming. Just like with
primes you have to try and see.
This rule is built into the fundamental nature of mathematics,
in the similar way that the speed of light appears to be an upper
boundary to the speed of moving objects. You can group by any
base you want, and it doesn’t affect the hidden chaotic structure
in numbers as given by primes and versatiles.
If you examine number theory in general, you find that it does
not have much of a structure. John Stillwell, Mathematics and
its History (p. 27): “number theory has never been submitted
to a systematic treatment like that undergone by elementary geometry
in Euclid’s Elements.”
So there is this sort of hidden chaos in numbers, with the primes
on one side and the versatiles on the other side.
Actually, our misplaced faith in the complete regularity of numbers
is somewhat a result of the base we use to group these numbers.
We can kind of get a feel for this chaos if we look at the following
chart. In the first three columns, we see the numbers, primes,
and versatiles in our HinduArabic baseten system. In the second
three columns, we see them in their purity, without the filterglassestool
of a basetengrouping system.
Primes  Numbers  Versatile 
  
p    v 
p    
  v  
p    
  v  
p    
  
  
  
p    
  v  
p    
  
  
  
p    
  
p    
  
  
  
p    
  v  
etc.  etc.  etc. 
With regard to versatile numbers,
there is a loose parallel to the history of psychology. Sigmund
Freud and others intently studied psychotic and neurotic behavior,
or aberration. This, of course, gave a skewed view of human nature.
Abraham Maslow decided to study healthy people or what he called
"selfactualized" people. This resulted in a new branch
of psychology called humanistic psychology. The two together provided
us with a more balanced view of humanity. In other words, I believe
that the general public has been taught about numbers with a minimum
of factors (primes), while they have not been taught about numbers
with a large number of factors (versatiles).
Versatile numbers may have potential use in the field of (w)holism.
Wholism can be defined as “the theory that whole entities,
as fundamental components of reality, have an existence other
than as the mere sum of their parts.” (Random House)
WHOLISM: the theory that whole entities, as fundamental components of reality, have an existence other than as the mere sum of their parts. (Random House Dict.) 
REDUCTIONISM: the theory that every complex phenomenon, especially in biology or psychology, can be explained by analyzing the simplest, most basic physical mechanisms that are in operation during the phenomennon. (Random House Dict.) 
Wholism has been attacked by
science philosopher Karl Popper and others. But Systems Analysis,
a practical application of wholism, has been used successfully
for many years. It was found that environmental problems did not
respond to compartmentalized attacks, but needed a larger viewpoint.
Harry J. White, in the Encyclopedia Americana, say Systems Analysis
is “an outgrowth of advanced technology,” because of
the “demands of society for more effective solutions of complex
environmental, production, and transportation problems.”
Also, as Bucky Fuller used to point out, when one considers the
alloy chromenickelsteel, it’s strength is not merely the
sum of its individual parts. The primary components are iron,
chromium, and nickel, with minor components of carbon, manganese,
and others. The tensile strength of iron is about 60,000 pounds
per square inch. Chromium is about 70,000 p.s.i.. Nickel is about
80,000 p.s.i. Carbon and the other minor components add another
50,000 p.s.i. Add them all up, and you get about 260,000 p.s.i.
But the tensile strength of chromenickelsteel runs to about
350,000 p.s.i.
The number 12, due to its highly versatile nature, due to the
structure of numbers, does appear to have an existence other than
as the mere sum of its parts. (12 = 11+1, 10+2, 9+3, 8+4, 7+5,
6+6.) The number has more factors than numbers up to twice its
sum.
Pure reductionists may say that we can still analyze why 12 is
a unique number, (because it has more factors than any smaller
number) but we must remember that versatile numbers, like primes,
are unpredictable. If pure reductionism were completely valid,
then we would expect that we could predict the next versatile
number.
For that matter, if pure reductionism were valid, then why couldn’t
we predict the next prime number? And since we can’t even
predict the next prime number, in the apparently abstract and
wellordered discipline of mathematics, can we expect a purely
reductionistic approach to work in the real world of science?
I am sort of glad that we have this unpredictability in numbers.
It seems to satisfy some inherent desire that the universe not
be completely predictable by science.
Philosophically, I have long since known that a completely predictable
universe would be a completely boring universe. We would have
no interest in it.
Douglas Hofstadter, in his notable work, Gödel, Escher, Bach,
also discusses “chaos in the midst of the most perfect, harmonious,
and beautiful of all creations: the system of natural numbers.”
(p. 398408)
Of course there is order in numbers. 1 + 1 + 1 + 1 + 1 + 1 + 1
+ 1 ... It is obvious and simple. Yet this simple order, so apparent
in counting and addition, somehow creates a sort of chaos in multiplication
and division.
PRIMES reductionistic elemental primitive barebones basic fundamental particle original primal rudimentary pure simple uncombined bedrock main key central bottom core crux underlying vital axiomatic necessary 
VERSATILES whole synergetic round systemic organic organizing omni sociable ecological cooperative versatile universal business economical practical convenient patterned symmetrical body well complete unbroken entire total 
At some points, where the versatiles
occur, there seems to be a sort of selforganization. One can
not help but wonder if this mirrors the selforganization that
occurs in living systems.
Perhaps we can generalize to: all things can be broken apart into
their basics, yet all things are a part of infinitely greater
structures.
Is the universe less a machine and more a living system or organism
(Newtonian paradigm to Gaia paradigm)? Should we begin to emphasize
less the fundamental (or prime) structure of things and more the
interconnectedness (versatility) of things?
The Fundamental Theorem of
Arithmetic is probably the foundation of what structure there
is in Number Theory. This states that every number can be broken
down into a different set of prime numbers.
However, opposed to this, but connected to it, we might also say
that every number is unique part of an infinite set of versatile
numbers. (Proof in Appendix E) Perhaps this should be called the Organic Theorem of
Arithmetic. On the one hand, each number can be broken down into
its parts, on the other hand each number is part of some greater
structure that is nested within still greater structures, ad infinitum.
(This theorem has been proven.)
Every number is nested within versatile numbers.
Versatile numbers, are numbers, in which as many different parts
as possible are interwoven into a whole, a unity. Unity + versatility
= Universal numbers.
Fundamental Theorem of Arithmetic 
Organic Theorem of Arithmetic 
Every
number can be broken down into a different set of primes.

Every
number is a part of an 
If you think about number from this perspective an interesting thing happens. The figure becomes the ground, and the ground becomes the figure.
Let’s imagine again the
human body (called by some a complex adaptive system) as an loose
analogy. It can be broken down chemically into its elements (primes):
carbon, oxygen, hydrogen, etc. These elements (primes) can be
combined in many different ways that are completely unremarkable
(nonversatile numbers), but when they are combined into certain
patterns, a human body (versatile number) results. No one could
have predicted it. But there it is. Perhaps it was inevitable.
The heart is a part of larger and larger systems. It is nested
within them. The circulatory system, the body, the family, the
community, the nation, all living things, the whole earth, the
solar system, etc. Sort of like the versatile number 8 is a part
of the versatile numbers 24, 48, 120, 240, 360, 720, 360, 840,
1680 and every remaining versatile number.
But the heart is not a part of the skeletal system, which is a
larger system than the heart. Sort of like the number 8 is not
part of every larger versatile number. It is not a part of 12
or 36 for example.
Is it a mere coincidence that carbon, the basis of all life as
we know it, is element number 6, with 6 protons, 6 neutrons, and
6 electrons? Perhaps. Is it a coincidence that Carbon 60, the
spherical molecule of great strength, has a versatile 60 number
of atoms? Perhaps. Is it a coincidence that the primary fuel of
the human body, C6H12O6, uses versatile numbers? Perhaps. Is it
a mere coincidence that DNA, the molecule that carries the blueprint
of our bodies uses a versatile 4 different parts? Perhaps.
An important class of numbers,
equal in importance to prime numbers, possibly known to the ancient
Babylonians, probably known about by Plato, was rediscovered by
Ramanujan 1915.
These numbers have more potential builtin patterns than any smaller
number. Numbers with no inherent patterns we call prime numbers
(except for 1).
Mathematicians, in the past, as well as in recent times, have
recommended using one of these versatile numbers, 12, for our
number base. However, thus far, society has stuck with the more
awkward base ten system.
In 1992, I independently developed a base 12 number system and
symbolism. In 1995, I independently rediscovered Ramanujan's “highly
composite numbers” and dubbed them "versatile numbers."
Things can be shared evenly easier with these numbers than with
other numbers. Thus, they have unexplored ramifications in economics,
business, psychology, and other social sciences.
OLD PARADIGM 
NEW PARADIGM 
reductionistic 
reductionisticwholistic 
prime 
primeversatile 
deficient 
nonversatile and versatile 
A specific name, versatile numbers, was coined, to allow the vast majority to understand the importance of these numbers, and to raise the status of these numbers to that of at least the status of “prime” numbers.
Reductionistic 
Wholistic 
ReductionisticWholistic 
prime 
nopattern 
prime 
composite 
patterned 
patterned 
highly composite 
organic 
versatile 
Currently almost 6 billion
people think, subconsciously, in terms of 10s and 100s, etc.,
not a very versatile system. Force a human mind to think in terms
of 10s and 100s, when the most versatile of all numbers are 2,
6, 12, 60, 360, and 2520, and you have perhaps to some degree
limited that mind, and possibly even warped that mind.
I sometimes try to imagine what would be the synergetic effect
if humankind’s collective numerical mind pattern were made
aware of versatile numbers, and were eventually brought into synchronization
with a versatile number system. For example, what would happen
if we had more versatile, “basetwelve filter glasses?”
Probably people would continue to fight with each other, no matter
what. However, numbers are a tool. Why not use the best possible
tool to lubricate Homo sapiens economic and social interaction?
1. Homo sapiens routinely uses
a mentaltoolfilterglassespattern to deal with continually
increasing amounts in the world around him or her.
2. The mentaltoolfilterglassespattern predominantly used on
Earth is probably  or ten.
3. Many amounts (versatile amounts) have relatively high potential
patterns builtin.
4. Homo sapiens might benefit from becoming more aware of these
more versatile amounts. (2, 4, 6, 12, 24, 36, 48, 60, 120, 180,
240, 360, 720,...)
5. Homo sapiens might benefit from using a mentaltoolfilterglasses
pattern with relatively high potential builtin patterns, as his
or her primarily mentaltoolfilterglassespattern for dealing
with the world around him or her.
Fundamental Thesis: Liberal use of versatile numbers (2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, ...) could lubricate a human’s interaction with surrounding humans and the surrounding universe, especially in an increasingly populated world.
Additional
minor theses:
a)
The unpredictability of prime numbers and versatile numbers suggests a
mathematical foundation for chaos in nature.
b)
The unpredictability of prime numbers and versatile numbers may oppose the idea
of a completely reductionistic approach to science.
c)
Versatile Numbers suggests a mathematical foundation for selforganization.
APPENDIX A: Sling Shot Method for determining some Versatile Numbers using Prime Numbers
This method was used by Ramanujan,
and was brought to my attention by Noam Shazeer and Matt Conroy.
(Note: Internet notation is used. * means “times”, ^
means “to the power of,” and _ means “subscript.”)
For any positive integer n,
let f(n) denote the number of factors of n. If n = (p_1^e_1)(p_2^e_2)...
(p_m^e_m) where p_i denotes the ith prime number, then the factors
of n will be numbers of the form (p_1^f_1)(p_2^f_2)...(p_m^f_m)
where 0<=f_i<=e_i for all i.
Since there are e_i +1 choices for each f_i, the number of factors
of n is (e_1+1)(e_2+1) ... (e_m +1)
To generate a versatile number, pick an arbitrary positive real
number r. Now let us find the value(s) of n which maximize (f(n)^r)/n.
If we find such an n, we can be sure that there is no x<n such
that f(x)>=f(n). If there were, (f(x)^r)/x would surely be
greater than (f(n)^r)/n. So n would be a versatile number.
Now let us find an n which maximizes (f(n)^r)/n.
Begin by noting that f(n)^r)/n = ((e_1+1)^r)/(p_1^e_1)*((e_2+1)^r)/(p_2^e_2)
... ((e_m+1)^r)/(p_m^e_m).
I will omit the proof that (f(n)^r)/n achieves a maximum. This
will become obvious in a minute.
To achieve our desired n, we must, for each i, pick an e_i which
maximizes ((e_i+1)^r/(p_i^e_i) > (((e_i1)+1^r)/(p_i^(e_i1))
if and only if
((e_i+1)/(ei))^r > p_i if and only if
((e_i+1)/(e_i)) > (p_i)^(1/r) if and only if
1+1/(e_i) > (p_i)^(1/r) if and only if
(e_i) > 1/((p_i)^(1/r))1)
So ((e_i+1)^r)/(p_i^e_i) is
clearly maximized when e_i is the greatest integer less than 1/((p_i)^(1/r)1).
This will be positive only when p_i < 2^r, i.e. only finitely
often.
So let us try our method,
Let r=4,
p_1e2, so e_1 =  1 / ((2) ^ (1/4)1))  = 5
p_2=3, so e_2 =  1 / ((3) ^ (1/4)1))  = 3
p_3=5, so e_3 =  1 / ((5) ^ (1/4)1))  = 2
when p_i = 7, 11, and 13, e_i = 1.
when p_1 > 16, e_i = 0
So, n=2^5 * 3^3 * 5^2 * 7 * 11 * 13 = 21621600
f(n) = 576
Let r=6
p_1=2, so e_1 =  1 / ((2) ^ (1/6)1))  = 8
p_2=3, so e_2 =  1 / ((3) ^ (1/6)1))  = 4
p_3=5, so e_3 =  1 / ((5) ^ (1/6)1))  = 3
p_4=7, so e_4 =  1 / ((7) ^ (1/6)1))  = 2
p_5=11, so e_5 = 1 / ((11) ^ (1/6)1))  = 2
when p_i = 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
e_i = 1
when p_i > 64, e_i = 0
So n=2^8 * 3^4 8 5^3 * 7^2 * 11^ 2 *
13*17*19*23*29*31*37*41*43*47*53*59*61
which has 30 digits.
f(n) = 13271040*
To produce a versatile number of about d digits, let r be approximately ln(d*ln(10))/ln(2). You should be able to compute the versatile number in about d*ln(d) time.
Note: this method does not produce all versatile numbers, e.g. no value of r produces the versatile number 4.
APPENDIX B: Teaching Versatile Numbers
First, one would want the students
to know how to factor a number. There are certain helpful rules
that are found in many mathematics textbooks, or which, I believe
could be derived by the students with proper guidance. Some examples
of these rules are: 1) all even numbers can be divided by two,
2) all numbers that end in 5 or 0 can be divided by five, 3) all
numbers whose digits add to a number divisible by 3 can be divided
by 3. There are other rules, and one could go into as much depth
as one wished.
Second, have the students use this knowledge of factoring to make
a factor table as shown at the beginning of this chapter. They
may need some assistance with some of the primes, but they should
be able to make a table up to say 60 or 120 (depending on their
grade level).
Third, have them circle all the primes. That is, have them circle
all the numbers with just two factors. One could then also go
into as much detail concerning primes as one wished. Last, have
them copy the table over (or use the same table), and have them
circle, in the last column, every time a number is greater than
all the previous numbers. In other words, (looking at the first
column now), every time a number had more factors than all the
previous numbers. Explain that these numbers are versatile numbers
and give an example of how they are more versatile than other
numbers, especially primes. The example of 23 students in a class
versus 24 students in a class might be a good example to use.
Of course, a more advanced class could use a computer program
to generate versatile numbers.
There may be better ways to teach this.
APPENDIX C: Possible TimeLine To Convert to a Versatile Number System
Of course, considering the chaotic nature of the universe, civilization, and technological growth, every social program should be tentative only, as it is quite possible that due to factors completely unknown to me, this entire paper would become irrelevant and obsolete.
Possible timeline to convert from a nonversatile to versatile economy:
2000 Begin to raise awareness
of general populace concerning versatile numbers.
2010 Versatile numbers taught in some textbooks alongside prime
numbers.
2050 Versatile numbers taught in all elementary, middle school,
and high school textbooks.
2100 Some individuals begin to use a versatile number system.
2200 Versatile number system spreads throughout society.
2400 Society has converted to a versatile number system including
all weights and measures, and monetary system.
APPENDIX
D:
Proof
of the Infinitude of Versatile Numbers
Consider a versatile number V1, and compute the next prime number P that
is greater than V1. Now consider the number V2 = PV1. The number V2 is obviously
greater than V1. More importantly, V2 has at least one more factor than V1,
namely P. The proof of this is simple: P cannot be a factor of V1 since V1 <
P, and all other factors of V1 must be factors of V2 because any factor F such
that FG = V1 would appear as F(GP) = V2.
Thus, since V2 has more factors than V1, the next versatile number after
V1 must be either V2 or appear before V2 numerically. This establishes the
existence of a versatile number that is greater than any given versatile number.
Inductively, this implies that the set of versatile numbers is unlimited
provided we have an infinitude of primes from which to draw P, which was
established by Euclid. Submitted by John Boyer (jboyer@uwi.com)
APPENDIX
E: Proof that every number, N, is a factor of some
versatile number, V.
Thm1:
(proved by Ramanujan, 1915): Let a[p][k] denote the exponent of the prime p in
the kth V number; For any fixed p, when k goes to infinity, a[p][k] tends to
infinity.
Thm
2: Let V be a versatile number with prime factorization
V
= PROD (i = 1 to inf, p_i ^ a_i).
Clearly,
a_i = 0 for sufficient i.
Now,
suppose a_i < a_(i+1). Then by
replacing p_i^a_i and p_(i+1)^a_(i+1)
with
p_i^a_(i+1) and p_(i+1)^a_i, respectively, in the product for V, we
obtain
a V' < V with the same number of factors.
This contradicts the fact that V
was
versatile. We conclude that a_i
>= a_(i+1), which is to say, the
exponents
in the prime factorization of V occur in descending order.
This
means
that the 0 exponents occur after the positive exponents.
Thm
3: Every positive number N is a
factor of some versatile number.
Proof:
Every finite number N has a prime factorization (fundamental theorem
of
arithmetic). Consider the highest
exponent E from the primes in N's
factorization,
and consider the highest prime P appearing in N's
factorization.
By Thm 1, there exists a versatile V[k] for which P^E is a
factor
(the power of P may be greater than E in V[k]). By Thm 2, every
prime
less than P appears in the factorization of V[k] with exponents of E
or
higher. Hence, N is a factor of V[k].
(by
J.L. Nicolas, John Boyer, and Dave Wilson.)
REFERENCES:
Apostle, Tom M., Introduction
to Analytic Number Theory, SpringerVerlag, NY, 1976.
Andrews, F. Emerson, “An Excursion in Numbers”, Atlantic
Monthly, Oct., 1934.
Asimov, Isaac, Realm of Numbers, Fawcett Crest, N.Y., 1967.
Dickson, L.E., History of the Theory of Numbers, 3 vols., Chelsea,
N.Y., 1952
Erdos, P., “On Highly Composite Numbers,” J. London
Mathematical Society, 19, 1944, pp.130133.
Frieberg, Richard, An Adverturer’s Guide to Number Theory,
McGrawHill, N.Y., 1968.
Hardy, G.W. Collected Papers of G.W. Hardy, Vol. VII, Clarendon
Press, Oxford, 1979.
Hofstadter, Douglas, Gödel, Echer, Bach: An Eternal Golden
Braid, Vintage Books, NY, 1989, 1979.
Hogan, James P., Mind Matters, Del Rey, 1997
Kanigel, Robert, The Man Who Knew Infinity, Charlesd Scribner’s
Sons, N.Y., 1991.
Lauirtzen, W.G., “Numbers of the Future” (video), Grassroots
Press, Glendale, CA, 1995.
Lauirtzen, W.G., Nature’s Numbers, booklet, Grassroots Press,
Glendale, CA, 1994.
Nicolas, JeanLouis, “On Highly Composite Numbers,”
Ramanujan Revisited, Academic Press, 1988.
Ore, Oystein, Number Theory and its History, McGrawHill, N.Y.,
1948.
Ramanujan, S. “Highly Composite Numbers,” Proceedings
of the London Mathematical
Society, Vol. 14, 1915.
Ratering, Steven, “An Interesting Subset of the Highly Composite
Numbers,” Mathematical Magazine, Vol 64, No. 5, Dec., 1991.
Robin, G. “Methods d’Optimisation pour un Probleme de
Theorie des Nombres,” R.A.I.R.O. Informatique theorique/Theoretical
Informatics, Vol. 17, No. 3., p. 239247, 1983.
Stillwell, John, Mathematics and its History, SpringerVerlag,
NY, 1989.
Thorne, Kip, Black Holes and Time Warps, Norton, NY, 1994.
Home  Articles  Email Lauritzen  Buy a Book  About Lauritzen 