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.

Babylon and Versatile Numbers

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 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 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 half-moon (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.

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 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 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 TOOL-PATTERNS IMPOSED ON CONTINUOUSLY INCREASING QUALITIES

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 inch--3 foot--1 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

Versatile Numbers
Part II

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" 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: 10because 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.

Versatile Numbers on the Internet

When I posted information on the internet regarding versatile 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 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.

Plato and Versatile 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 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 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 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."

Ramanujan and Versatile 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 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."

Nomenclature

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.

Paul Erdös

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 Jean-Louis 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 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 versatile numbers or numbers with relatively large numbers of factors.

Five Platonic Solids corners or vertices edges faces

name	corners	edges	faces
tetrahedron (4-nook)	4	6	4
octahedron (6-nook)	6	12	8
cube (8-nook)	8	12	6
icosahedron (12-nook)	12	30	20
docedahedron (20-nook)	30	20	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.

Primordial Numbers

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 "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 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, ... (no lower number has as many prime factors)

Versatile Numbers: 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, ... (no lower number has as many factors)

Applications of Versatile Numbers

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.

Grouping by Dozens

.

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

twelve-pack

six-pack

twelve parts to the chromatic scale

twenty-four 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 alphabe

twelve, twenty-four, and thirty-six 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 so-called scientific base ten metric system. For example, in 1971, the British switched from half-pennies, pennies, threepence, sixpence, shillings, half-crowns, 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 non-versatile, 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 non-versatile 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.

Base Ten Glasses

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, 6-packs, 12-packs, 12-24-36 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.

Various Ways to Mentally Group Numbers -- The Base of Various Number Systems

Hindu-Arabic (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 | + | + | + | + | ...).

NUMBERS AND THEIR PATTERNS

| no patterns || no patterns (prime) ||| no patterns (prime) |||| = || || (versatile) ||||| no patterns (prime) |||||| = ||| ||| = || || || (versatile) ||||||| no patterns (prime) |||||||| = |||| |||| = || || || || ||||||||| = ||| ||| |||||||||| = ||||| ||||| = || || || || || ||||||||||| no patterns (prime) |||||||||||| = |||||| |||||| = |||| |||| |||| = ||| ||| ||| ||| = || || || || || || (versatile) etc.

(versatile numbers have more possible patterns than any smaller number)

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, “base-ten 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 base-twelve-filter-glasses would say, “I see twelve and twelve, or two dozen.”

THE FUNDAMENTAL PATTERNS OF A NUMBER

|||||||||||| Twelve as a pure amount or number without base-10 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 base-ten-tool-filter-glasses 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 base-twelve-filter-glasses would have immediately shown the versatile number 24. Practical filter-glasses 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 non-versatile 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 0-9 but entirely different symbols of my own invention) derived from versatile numbers. I have personally taught this system to over 6000 students.

The Gravitational Symmetry Limit

In my video of 1995, I relate the versatile number 12 to easily remembered geometrical models--models 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 “turn-same.” 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 turn-same of 60. The symmetries of the two balls, three balls, four balls, six balls, and twelve balls are 4, 6, 12, 24, and 60--all 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 built-in patterns. Patterns or repetitions are what make symmetry.

end Part II of III

Versatile Numbers
Part III
The Largest Versatiles

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 e-mail 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.

The Hidden Chaotic Structure in Numbers

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 Hindu-Arabic base-ten system. In the second three columns, we see them in their purity, without the filter-glasses-tool of a base-ten-grouping system.

Primes	Numbers	Versatile
	|
p	||	v
p	|||
	||||	v
p	|||||
	||||||	v
p	|||||||
	||||||||
	|||||||||
	||||||||||
p	|||||||||||
	||||||||||||	v
p	|||||||||||||
	||||||||||||||
	|||||||||||||||
	||||||||||||||||
p	|||||||||||||||||
	||||||||||||||||||
p	|||||||||||||||||||
	||||||||||||||||||||
	|||||||||||||||||||||
	||||||||||||||||||||||
p	|||||||||||||||||||||||
	||||||||||||||||||||||||	v
etc.	etc.	etc.

Synergy, Wholism, and Systems Theory

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 "self-actualized" 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 chrome-nickel-steel, 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 chrome-nickel-steel 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 well-ordered 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. 398-408)

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 eductionistic elemental primitive bare-bones 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 self-organization. One can not help but wonder if this mirrors the self-organization 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

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 = Uni-versal numbers.

Fundamental Theorem of Arithmetic	Organic Theorem of Arithmetic
Every number can be broken down into a different set of primes. Example: 8=2x2x2	Every number is a part of aninfinite set of versatiles. Example: 8 is a part of 24, 48, 120, 240, ...

If you think about number from this perspective an interesting thing happens. The figure becomes the ground, and the ground becomes the figure.

Versatile Life Systems

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 (non-versatile 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.

Summary

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 built-in 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	reductionistic-wholistic
prime	prime-versatile
deficient perfect abundant	non-versatile 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.

POSSIBLE NUMBER NOMENCLATURES

Reductionistic	Wholistic	Reductionistic-Wholistic
prime	no-pattern	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, “base-twelve 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?

Conclusions

. Homo sapiens routinely uses a mental-tool-filter-glasses-pattern to deal with continually increasing amounts in the world around him or her.
2. The mental-tool-filter-glasses-pattern predominantly used on Earth is probably |||||||||| or ten.
3. Many amounts (versatile amounts) have relatively high potential patterns built-in.
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 mental-tool-filter-glasses pattern with relatively high potential built-in patterns, as his or her primarily mental-tool-filter-glasses-pattern 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 self-organization.

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=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_i-1)+1^r)/(p_i^(e_i-1)) if and only if
((e_i+1)/(e-i))^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 Time-Line 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 time-line to convert from a non-versatile 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 k-th 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 afactor (the power of P may be greater than E in V[k]). By Thm 2, everyprime less than P appears in the factorization of V[k] with exponents of Eor higher. Hence, N is a factor of V[k].(by J.-L. Nicolas, John Boyer, and Dave Wilson.)

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